P. 373 of Theorem List - Metamath Proof Explorer

Type

Label

Description

Statement

Theorem

3impcombi 37201

A 1-hypothesis propositional calculus deduction. (Contributed by Alan Sare, 25-Sep-2017.)

|- ( ( ph /\ ps /\ ph ) -> ( ch <-> th ) )   =>    |- ( ( ps /\ ph /\ ch ) -> th )

Theorem

3imp231 37202

Importation inference. (Contributed by Alan Sare, 17-Oct-2017.)

|- ( ph -> ( ps -> ( ch -> th ) ) )   =>    |- ( ( ps /\ ch /\ ph ) -> th )

21.29.6  Theorems proved using Virtual Deduction

Theorem

trsspwALT 37203

Virtual deduction proof of the left-to-right implication of dftr4 4474. A transitive class is a subset of its power class. This proof corresponds to the virtual deduction proof of dftr4 4474 without accumulating results. (Contributed by Alan Sare, 29-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( Tr A -> A C_ ~P A )

Theorem

trsspwALT2 37204

Virtual deduction proof of trsspwALT 37203. This proof is the same as the proof of trsspwALT 37203 except each virtual deduction symbol is replaced by its non-virtual deduction symbol equivalent. A transitive class is a subset of its power class. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( Tr A -> A C_ ~P A )

Theorem

trsspwALT3 37205

Short predicate calculus proof of the left-to-right implication of dftr4 4474. A transitive class is a subset of its power class. This proof was constructed by applying Metamath's minimize command to the proof of trsspwALT2 37204, which is the virtual deduction proof trsspwALT 37203 without virtual deductions. (Contributed by Alan Sare, 30-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( Tr A -> A C_ ~P A )

Theorem

sspwtr 37206

Virtual deduction proof of the right-to-left implication of dftr4 4474. A class which is a subclass of its power class is transitive. This proof corresponds to the virtual deduction proof of sspwtr 37206 without accumulating results. (Contributed by Alan Sare, 2-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( A C_ ~P A -> Tr A )

Theorem

sspwtrALT 37207

Virtual deduction proof of sspwtr 37206. This proof is the same as the proof of sspwtr 37206 except each virtual deduction symbol is replaced by its non-virtual deduction symbol equivalent. A class which is a subclass of its power class is transitive. (Contributed by Alan Sare, 3-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( A C_ ~P A -> Tr A )

Theorem

csbabgOLD 37208*

Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) Obsolete as of 19-Aug-2018. Use csbab 3765 instead. (New usage is discouraged.) (Proof modification is discouraged.)

|- ( A e. V -> [_ A / x ]_ { y | ph } = { y | [. A / x ]. ph } )

Theorem

csbunigOLD 37209

Distribute proper substitution through the union of a class. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 22-Aug-2018. Use csbuni 4196 instead. (New usage is discouraged.) (Proof modification is discouraged.)

|- ( A e. V -> [_ A / x ]_ U. B = U. [_ A / x ]_ B )

Theorem

csbfv12gALTOLD 37210

Move class substitution in and out of a function value. The proof is derived from the virtual deduction proof csbfv12gALTVD 37293. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 20-Aug-2018. Use csbfv12 5883 instead. (Proof modification is discouraged.) (New usage is discouraged.)

|- ( A e. C -> [_ A / x ]_ ( F ` B ) = ( [_ A / x ]_ F ` [_ A / x ]_ B ) )

Theorem

csbxpgOLD 37211

Distribute proper substitution through the Cartesian product of two classes. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 23-Aug-2018. Use csbrn 5276 instead. (New usage is discouraged.) (Proof modification is discouraged.)

|- ( A e. D -> [_ A / x ]_ ( B X. C ) = ( [_ A / x ]_ B X. [_ A / x ]_ C ) )

Theorem

csbingOLD 37212

Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.) Obsolete as of 18-Aug-2018. Use csbin 3767 instead. (New usage is discouraged.) (Proof modification is discouraged.)

|- ( A e. B -> [_ A / x ]_ ( C i^i D ) = ( [_ A / x ]_ C i^i [_ A / x ]_ D ) )

Theorem

csbresgOLD 37213

Distribute proper substitution through the restriction of a class. csbresgOLD 37213 is derived from the virtual deduction proof csbresgVD 37289. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 23-Aug-2018. Use csbres 5086 instead. (New usage is discouraged.) (Proof modification is discouraged.)

|- ( A e. V -> [_ A / x ]_ ( B |` C ) = ( [_ A / x ]_ B |` [_ A / x ]_ C ) )

Theorem

csbrngOLD 37214

Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 23-Aug-2018. Use csbrn 5276 instead. (New usage is discouraged.) (Proof modification is discouraged.)

|- ( A e. V -> [_ A / x ]_ ran B = ran [_ A / x ]_ B )

Theorem

csbima12gALTOLD 37215

Move class substitution in and out of the image of a function. The proof is derived from the virtual deduction proof csbima12gALTVD 37291. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 20-Aug-2018. Use csbfv12 5883 instead. (Proof modification is discouraged.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( A e. C -> [_ A / x ]_ ( F " B ) = ( [_ A / x ]_ F " [_ A / x ]_ B ) )

Theorem

sspwtrALT2 37216

Short predicate calculus proof of the right-to-left implication of dftr4 4474. A class which is a subclass of its power class is transitive. This proof was constructed by applying Metamath's minimize command to the proof of sspwtrALT 37207, which is the virtual deduction proof sspwtr 37206 without virtual deductions. (Contributed by Alan Sare, 3-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( A C_ ~P A -> Tr A )

Theorem

pwtrVD 37217

Virtual deduction proof of pwtr 4626; see pwtrrVD 37218 for the converse. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( Tr A -> Tr ~P A )

Theorem

pwtrrVD 37218

Virtual deduction proof of pwtr 4626; see pwtrVD 37217 for the converse. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- A e. _V   =>    |- ( Tr ~P A -> Tr A )

Theorem

suctrALT 37219

The successor of a transitive class is transitive. The proof of http://us.metamath.org/other/completeusersproof/suctrvd.html is a Virtual Deduction proof verified by automatically transforming it into the Metamath proof of suctrALT 37219 using completeusersproof, which is verified by the Metamath program. The proof of http://us.metamath.org/other/completeusersproof/suctrro.html is a form of the completed proof which preserves the Virtual Deduction proof's step numbers and their ordering. See suctr 5485 for the original proof. (Contributed by Alan Sare, 11-Apr-2009.) (Revised by Alan Sare, 12-Jun-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( Tr A -> Tr suc A )

Theorem

snssiALTVD 37220

Virtual deduction proof of snssiALT 37221. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( A e. B -> { A } C_ B )

Theorem

snssiALT 37221

If a class is an element of another class, then its singleton is a subclass of that other class. Alternate proof of snssi 4085. This theorem was automatically generated from snssiALTVD 37220 using a translation program. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( A e. B -> { A } C_ B )

Theorem

snsslVD 37222

Virtual deduction proof of snssl 37223. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- A e. _V   =>    |- ( { A } C_ B -> A e. B )

Theorem

snssl 37223

If a singleton is a subclass of another class, then the singleton's element is an element of that other class. This theorem is the right-to-left implication of the biconditional snss 4065. The proof of this theorem was automatically generated from snsslVD 37222 using a tools command file, translateMWO.cmd, by translating the proof into its non-virtual deduction form and minimizing it. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- A e. _V   =>    |- ( { A } C_ B -> A e. B )

Theorem

snelpwrVD 37224

Virtual deduction proof of snelpwi 4618. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( A e. B -> { A } e. ~P B )

Theorem

unipwrVD 37225

Virtual deduction proof of unipwr 37226. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- A C_ U. ~P A

Theorem

unipwr 37226

A class is a subclass of the union of its power class. This theorem is the right-to-left subclass lemma of unipw 4623. The proof of this theorem was automatically generated from unipwrVD 37225 using a tools command file , translateMWO.cmd , by translating the proof into its non-virtual deduction form and minimizing it. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- A C_ U. ~P A

Theorem

sstrALT2VD 37227

Virtual deduction proof of sstrALT2 37228. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( ( A C_ B /\ B C_ C ) -> A C_ C )

Theorem

sstrALT2 37228

Virtual deduction proof of sstr 3408, transitivity of subclasses, Theorem 6 of [Suppes] p. 23. This theorem was automatically generated from sstrALT2VD 37227 using the command file translate_without_overwriting.cmd . It was not minimized because the automated minimization excluding duplicates generates a minimized proof which, although not directly containing any duplicates, indirectly contains a duplicate. That is, the trace back of the minimized proof contains a duplicate. This is undesirable because some step(s) of the minimized proof use the proven theorem. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( ( A C_ B /\ B C_ C ) -> A C_ C )

Theorem

suctrALT2VD 37229

Virtual deduction proof of suctrALT2 37230. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( Tr A -> Tr suc A )

Theorem

suctrALT2 37230

Virtual deduction proof of suctr 5485. The sucessor of a transitive class is transitive. This proof was generated automatically from the virtual deduction proof suctrALT2VD 37229 using the tools command file translate_without_overwriting_minimize_excluding_duplicates.cmd . (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( Tr A -> Tr suc A )

Theorem

elex2VD 37231*

Virtual deduction proof of elex2 3026. (Contributed by Alan Sare, 25-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( A e. B -> E. x x e. B )

Theorem

elex22VD 37232*

Virtual deduction proof of elex22 3027. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( ( A e. B /\ A e. C ) -> E. x ( x e. B /\ x e. C ) )

Theorem

eqsbc3rVD 37233*

Virtual deduction proof of eqsbc3r 3292. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( A e. B -> ( [. A / x ]. C = x <-> C = A ) )

Theorem

zfregs2VD 37234*

Virtual deduction proof of zfregs2 8204. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( A =/= (/) -> -. A. x e. A E. y ( y e. A /\ y e. x ) )

Theorem

tpid3gVD 37235

Virtual deduction proof of tpid3g 4056. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( A e. B -> A e. { C , D , A } )

Theorem

en3lplem1VD 37236*

Virtual deduction proof of en3lplem1 8106. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( ( A e. B /\ B e. C /\ C e. A ) -> ( x = A -> E. y ( y e. { A , B , C } /\ y e. x ) ) )

Theorem

en3lplem2VD 37237*

Virtual deduction proof of en3lplem2 8107. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( ( A e. B /\ B e. C /\ C e. A ) -> ( x e. { A , B , C } -> E. y ( y e. { A , B , C } /\ y e. x ) ) )

Theorem

en3lpVD 37238

Virtual deduction proof of en3lp 8108. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- -. ( A e. B /\ B e. C /\ C e. A )

21.29.7  Theorems proved using Virtual Deduction with mmj2 assistance

Theorem

simplbi2VD 37239

Virtual deduction proof of simplbi2 635. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

h1::	|- ( ph <-> ( ps /\ ch ) )
3:1,?: e0a 37156	|- ( ( ps /\ ch ) -> ph )
qed:3,?: e0a 37156	|- ( ps -> ( ch -> ph ) )

The proof of simplbi2 635 was automatically derived from it. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( ph <-> ( ps /\ ch ) )   =>    |- ( ps -> ( ch -> ph ) )

Theorem

3ornot23VD 37240

Virtual deduction proof of 3ornot23 36866. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

1::

|- (. ( -. ph /\ -. ps ) ->. ( -. ph /\ -. ps ) ).
2::	|- (. ( -. ph /\ -. ps ) ,. ( ch \/ ph \/ ps ) ->. ( ch \/ ph \/ ps ) ).
3:1,?: e1a 37006	|- (. ( -. ph /\ -. ps ) ->. -. ph ).
4:1,?: e1a 37006	|- (. ( -. ph /\ -. ps ) ->. -. ps ).
5:3,4,?: e11 37067	|- (. ( -. ph /\ -. ps ) ->. -. ( ph \/ ps ) ).
6:2,?: e2 37010	|- (. ( -. ph /\ -. ps ) ,. ( ch \/ ph \/ ps ) ->. ( ch \/ ( ph \/ ps ) ) ).
7:5,6,?: e12 37108	|- (. ( -. ph /\ -. ps ) ,. ( ch \/ ph \/ ps ) ->. ch ).
8:7:	|- (. ( -. ph /\ -. ps ) ->. ( ( ch \/ ph \/ ps ) -> ch ) ).
qed:8:	|- ( ( -. ph /\ -. ps ) -> ( ( ch \/ ph \/ ps ) -> ch ) )

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( ( -. ph /\ -. ps ) -> ( ( ch \/ ph \/ ps ) -> ch ) )

Theorem

orbi1rVD 37241

Virtual deduction proof of orbi1r 36867. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

1::	|- (. ( ph <-> ps ) ->. ( ph <-> ps ) ).
2::	|- (. ( ph <-> ps ) ,. ( ch \/ ph ) ->. ( ch \/ ph ) ).
3:2,?: e2 37010	|- (. ( ph <-> ps ) ,. ( ch \/ ph ) ->. ( ph \/ ch ) ).
4:1,3,?: e12 37108	|- (. ( ph <-> ps ) ,. ( ch \/ ph ) ->. ( ps \/ ch ) ).
5:4,?: e2 37010	|- (. ( ph <-> ps ) ,. ( ch \/ ph ) ->. ( ch \/ ps ) ).
6:5:	|- (. ( ph <-> ps ) ->. ( ( ch \/ ph ) -> ( ch \/ ps ) ) ).
7::	|- (. ( ph <-> ps ) ,. ( ch \/ ps ) ->. ( ch \/ ps ) ).
8:7,?: e2 37010	|- (. ( ph <-> ps ) ,. ( ch \/ ps ) ->. ( ps \/ ch ) ).
9:1,8,?: e12 37108	|- (. ( ph <-> ps ) ,. ( ch \/ ps ) ->. ( ph \/ ch ) ).
10:9,?: e2 37010	|- (. ( ph <-> ps ) ,. ( ch \/ ps ) ->. ( ch \/ ph ) ).
11:10:	|- (. ( ph <-> ps ) ->. ( ( ch \/ ps ) -> ( ch \/ ph ) ) ).
12:6,11,?: e11 37067	|- (. ( ph <-> ps ) ->. ( ( ch \/ ph ) <-> ( ch \/ ps ) ) ).
qed:12:	|- ( ( ph <-> ps ) -> ( ( ch \/ ph ) <-> ( ch \/ ps ) ) )

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( ( ph <-> ps ) -> ( ( ch \/ ph ) <-> ( ch \/ ps ) ) )

Theorem

bitr3VD 37242

Virtual deduction proof of bitr3 36868. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

1::	|- (. ( ph <-> ps ) ->. ( ph <-> ps ) ).
2:1,?: e1a 37006	|- (. ( ph <-> ps ) ->. ( ps <-> ph ) ).
3::	|- (. ( ph <-> ps ) ,. ( ph <-> ch ) ->. ( ph <-> ch ) ).
4:3,?: e2 37010	|- (. ( ph <-> ps ) ,. ( ph <-> ch ) ->. ( ch <-> ph ) ).
5:2,4,?: e12 37108	|- (. ( ph <-> ps ) ,. ( ph <-> ch ) ->. ( ps <-> ch ) ).
6:5:	|- (. ( ph <-> ps ) ->. ( ( ph <-> ch ) -> ( ps <-> ch ) ) ).
qed:6:	|- ( ( ph <-> ps ) -> ( ( ph <-> ch ) -> ( ps <-> ch ) ) )

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( ( ph <-> ps ) -> ( ( ph <-> ch ) -> ( ps <-> ch ) ) )

Theorem

3orbi123VD 37243

Virtual deduction proof of 3orbi123 36869. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

1::	|- (. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ->. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ).
2:1,?: e1a 37006	|- (. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ->. ( ph <-> ps ) ).
3:1,?: e1a 37006	|- (. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ->. ( ch <-> th ) ).
4:1,?: e1a 37006	|- (. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ->. ( ta <-> et ) ).
5:2,3,?: e11 37067	|- (. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ->. ( ( ph \/ ch ) <-> ( ps \/ th ) ) ).
6:5,4,?: e11 37067	|- (. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ->. ( ( ( ph \/ ch ) \/ ta ) <-> ( ( ps \/ th ) \/ et ) ) ).
7:?:	|- ( ( ( ph \/ ch ) \/ ta ) <-> ( ph \/ ch \/ ta ) )
8:6,7,?: e10 37073	|- (. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ->. ( ( ph \/ ch \/ ta ) <-> ( ( ps \/ th ) \/ et ) ) ).
9:?:	|- ( ( ( ps \/ th ) \/ et ) <-> ( ps \/ th \/ et ) )
10:8,9,?: e10 37073	|- (. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ->. ( ( ph \/ ch \/ ta ) <-> ( ps \/ th \/ et ) ) ).
qed:10:	|- ( ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) -> ( ( ph \/ ch \/ ta ) <-> ( ps \/ th \/ et ) ) )

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) -> ( ( ph \/ ch \/ ta ) <-> ( ps \/ th \/ et ) ) )

Theorem

sbc3orgVD 37244

Virtual deduction proof of sbc3orgOLD 36894. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

1::	|- (. A e. B ->. A e. B ).
2:1,?: e1a 37006	|- (. A e. B ->. ( [. A / x ]. ( ( ph \/ ps ) \/ ch ) <-> ( [. A / x ]. ( ph \/ ps ) \/ [. A / x ]. ch ) ) ).
3::	|- ( ( ( ph \/ ps ) \/ ch ) <-> ( ph \/ ps \/ ch ) )
32:3:	|- A. x ( ( ( ph \/ ps ) \/ ch ) <-> ( ph \/ ps \/ ch ) )
33:1,32,?: e10 37073	|- (. A e. B ->. [. A / x ]. ( ( ( ph \/ ps ) \/ ch ) <-> ( ph \/ ps \/ ch ) ) ).
4:1,33,?: e11 37067	|- (. A e. B ->. ( [. A / x ]. ( ( ph \/ ps ) \/ ch ) <-> [. A / x ]. ( ph \/ ps \/ ch ) ) ).
5:2,4,?: e11 37067	|- (. A e. B ->. ( [. A / x ]. ( ph \/ ps \/ ch ) <-> ( [. A / x ]. ( ph \/ ps ) \/ [. A / x ]. ch ) ) ).
6:1,?: e1a 37006	|- (. A e. B ->. ( [. A / x ]. ( ph \/ ps ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps ) ) ).
7:6,?: e1a 37006	|- (. A e. B ->. ( ( [. A / x ]. ( ph \/ ps ) \/ [. A / x ]. ch ) <-> ( ( [. A / x ]. ph \/ [. A / x ]. ps ) \/ [. A / x ]. ch ) ) ).
8:5,7,?: e11 37067	|- (. A e. B ->. ( [. A / x ]. ( ph \/ ps \/ ch ) <-> ( ( [. A / x ]. ph \/ [. A / x ]. ps ) \/ [. A / x ]. ch ) ) ).
9:?:	|- ( ( ( [. A / x ]. ph \/ [. A / x ]. ps ) \/ [. A / x ]. ch ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps \/ [. A / x ]. ch ) )
10:8,9,?: e10 37073	|- (. A e. B ->. ( [. A / x ]. ( ph \/ ps \/ ch ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps \/ [. A / x ]. ch ) ) ).
qed:10:	|- ( A e. B -> ( [. A / x ]. ( ph \/ ps \/ ch ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps \/ [. A / x ]. ch ) ) )

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( A e. B -> ( [. A / x ]. ( ph \/ ps \/ ch ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps \/ [. A / x ]. ch ) ) )

Theorem

19.21a3con13vVD 37245*

Virtual deduction proof of alrim3con13v 36895. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

1::	|- (. ( ph -> A. x ph ) ->. ( ph -> A. x ph ) ).
2::	|- (. ( ph -> A. x ph ) ,. ( ps /\ ph /\ ch ) ->. ( ps /\ ph /\ ch ) ).
3:2,?: e2 37010	|- (. ( ph -> A. x ph ) ,. ( ps /\ ph /\ ch ) ->. ps ).
4:2,?: e2 37010	|- (. ( ph -> A. x ph ) ,. ( ps /\ ph /\ ch ) ->. ph ).
5:2,?: e2 37010	|- (. ( ph -> A. x ph ) ,. ( ps /\ ph /\ ch ) ->. ch ).
6:1,4,?: e12 37108	|- (. ( ph -> A. x ph ) ,. ( ps /\ ph /\ ch ) ->. A. x ph ).
7:3,?: e2 37010	|- (. ( ph -> A. x ph ) ,. ( ps /\ ph /\ ch ) ->. A. x ps ).
8:5,?: e2 37010	|- (. ( ph -> A. x ph ) ,. ( ps /\ ph /\ ch ) ->. A. x ch ).
9:7,6,8,?: e222 37015	|- (. ( ph -> A. x ph ) ,. ( ps /\ ph /\ ch ) ->. ( A. x ps /\ A. x ph /\ A. x ch ) ).
10:9,?: e2 37010	|- (. ( ph -> A. x ph ) ,. ( ps /\ ph /\ ch ) ->. A. x ( ps /\ ph /\ ch ) ).
11:10:in2	|- (. ( ph -> A. x ph ) ->. ( ( ps /\ ph /\ ch ) -> A. x ( ps /\ ph /\ ch ) ) ).
qed:11:in1	|- ( ( ph -> A. x ph ) -> ( ( ps /\ ph /\ ch ) -> A. x ( ps /\ ph /\ ch ) ) )

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( ( ph -> A. x ph ) -> ( ( ps /\ ph /\ ch ) -> A. x ( ps /\ ph /\ ch ) ) )

Theorem

exbirVD 37246

Virtual deduction proof of exbir 36834. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

1::	|- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) ->. ( ( ph /\ ps ) -> ( ch <-> th ) ) ).
2::	|- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) ,. ( ph /\ ps ) ->. ( ph /\ ps ) ).
3::	|- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) ,. ( ph /\ ps ) , th ->. th ).
5:1,2,?: e12 37108	|- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) , ( ph /\ ps ) ->. ( ch <-> th ) ).
6:3,5,?: e32 37142	|- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) , ( ph /\ ps ) , th ->. ch ).
7:6:	|- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) , ( ph /\ ps ) ->. ( th -> ch ) ).
8:7:	|- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) ->. ( ( ph /\ ps ) -> ( th -> ch ) ) ).
9:8,?: e1a 37006	|- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) ->. ( ph -> ( ps -> ( th -> ch ) ) ) ).
qed:9:	|- ( ( ( ph /\ ps ) -> ( ch <-> th ) ) -> ( ph -> ( ps -> ( th -> ch ) ) ) )

(Contributed by Alan Sare, 13-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( ( ( ph /\ ps ) -> ( ch <-> th ) ) -> ( ph -> ( ps -> ( th -> ch ) ) ) )

Theorem

exbiriVD 37247

Virtual deduction proof of exbiri 632. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

h1::	|- ( ( ph /\ ps ) -> ( ch <-> th ) )
2::	|- (. ph ->. ph ).
3::	|- (. ph ,. ps ->. ps ).
4::	|- (. ph ,. ps ,. th ->. th ).
5:2,1,?: e10 37073	|- (. ph ->. ( ps -> ( ch <-> th ) ) ).
6:3,5,?: e21 37114	|- (. ph ,. ps ->. ( ch <-> th ) ).
7:4,6,?: e32 37142	|- (. ph ,. ps ,. th ->. ch ).
8:7:	|- (. ph ,. ps ->. ( th -> ch ) ).
9:8:	|- (. ph ->. ( ps -> ( th -> ch ) ) ).
qed:9:	|- ( ph -> ( ps -> ( th -> ch ) ) )

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( ( ph /\ ps ) -> ( ch <-> th ) )   =>    |- ( ph -> ( ps -> ( th -> ch ) ) )

Theorem

rspsbc2VD 37248*

Virtual deduction proof of rspsbc2 36896. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

1::	|- (. A e. B ->. A e. B ).
2::	|- (. A e. B ,. C e. D ->. C e. D ).
3::	|- (. A e. B ,. C e. D ,. A. x e. B A. y e. D ph ->. A. x e. B A. y e. D ph ).
4:1,3,?: e13 37132	|- (. A e. B ,. C e. D ,. A. x e. B A. y e. D ph ->. [. A / x ]. A. y e. D ph ).
5:1,4,?: e13 37132	|- (. A e. B ,. C e. D ,. A. x e. B A. y e. D ph ->. A. y e. D [. A / x ]. ph ).
6:2,5,?: e23 37139	|- (. A e. B ,. C e. D ,. A. x e. B A. y e. D ph ->. [. C / y ]. [. A / x ]. ph ).
7:6:	|- (. A e. B ,. C e. D ->. ( A. x e. B A. y e. D ph -> [. C / y ]. [. A / x ]. ph ) ).
8:7:	|- (. A e. B ->. ( C e. D -> ( A. x e. B A. y e. D ph -> [. C / y ]. [. A / x ]. ph ) ) ).
qed:8:	|- ( A e. B -> ( C e. D -> ( A. x e. B A. y e. D ph -> [. C / y ]. [. A / x ]. ph ) ) )

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( A e. B -> ( C e. D -> ( A. x e. B A. y e. D ph -> [. C / y ]. [. A / x ]. ph ) ) )

Theorem

3impexpVD 37249

Virtual deduction proof of 3impexp 1234. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

1::	|- (. ( ( ph /\ ps /\ ch ) -> th ) ->. ( ( ph /\ ps /\ ch ) -> th ) ).
2::	|- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) )
3:1,2,?: e10 37073	|- (. ( ( ph /\ ps /\ ch ) -> th ) ->. ( ( ( ph /\ ps ) /\ ch ) -> th ) ).
4:3,?: e1a 37006	|- (. ( ( ph /\ ps /\ ch ) -> th ) ->. ( ( ph /\ ps ) -> ( ch -> th ) ) ).
5:4,?: e1a 37006	|- (. ( ( ph /\ ps /\ ch ) -> th ) ->. ( ph -> ( ps -> ( ch -> th ) ) ) ).
6:5:	|- ( ( ( ph /\ ps /\ ch ) -> th ) -> ( ph -> ( ps -> ( ch -> th ) ) ) )
7::	|- (. ( ph -> ( ps -> ( ch -> th ) ) ) ->. ( ph -> ( ps -> ( ch -> th ) ) ) ).
8:7,?: e1a 37006	|- (. ( ph -> ( ps -> ( ch -> th ) ) ) ->. ( ( ph /\ ps ) -> ( ch -> th ) ) ).
9:8,?: e1a 37006	|- (. ( ph -> ( ps -> ( ch -> th ) ) ) ->. ( ( ( ph /\ ps ) /\ ch ) -> th ) ).
10:2,9,?: e01 37070	|- (. ( ph -> ( ps -> ( ch -> th ) ) ) ->. ( ( ph /\ ps /\ ch ) -> th ) ).
11:10:	|- ( ( ph -> ( ps -> ( ch -> th ) ) ) -> ( ( ph /\ ps /\ ch ) -> th ) )
qed:6,11,?: e00 37152	|- ( ( ( ph /\ ps /\ ch ) -> th ) <-> ( ph -> ( ps -> ( ch -> th ) ) ) )

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( ( ( ph /\ ps /\ ch ) -> th ) <-> ( ph -> ( ps -> ( ch -> th ) ) ) )

Theorem

3impexpbicomVD 37250

Virtual deduction proof of 3impexpbicom 36835. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

1::	|- (. ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) ->. ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) ).
2::	|- ( ( th <-> ta ) <-> ( ta <-> th ) )
3:1,2,?: e10 37073	|- (. ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) ->. ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) ).
4:3,?: e1a 37006	|- (. ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) ->. ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ).
5:4:	|- ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) -> ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) )
6::	|- (. ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ->. ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ).
7:6,?: e1a 37006	|- (. ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ->. ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) ).
8:7,2,?: e10 37073	|- (. ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ->. ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) ).
9:8:	|- ( ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) -> ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) )
qed:5,9,?: e00 37152	|- ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) <-> ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) )

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) <-> ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) )

Theorem

3impexpbicomiVD 37251

Virtual deduction proof of 3impexpbicomi 36836. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

h1::	|- ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) )
qed:1,?: e0a 37156	|- ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) )

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) )   =>    |- ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) )

Theorem

sbcel1gvOLD 37252*

Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) Obsolete as of 17-Aug-2018. Use sbcel1v 3294 instead. (New usage is discouraged.) (Proof modification is discouraged.)

|- ( A e. V -> ( [. A / x ]. x e. B <-> A e. B ) )

Theorem

sbcoreleleqVD 37253*

Virtual deduction proof of sbcoreleleq 36897. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

1::	|- (. A e. B ->. A e. B ).
2:1,?: e1a 37006	|- (. A e. B ->. ( [. A / y ]. x e. y <-> x e. A ) ).
3:1,?: e1a 37006	|- (. A e. B ->. ( [. A / y ]. y e. x <-> A e. x ) ).
4:1,?: e1a 37006	|- (. A e. B ->. ( [. A / y ]. x = y <-> x = A ) ).
5:2,3,4,?: e111 37053	|- (. A e. B ->. ( ( x e. A \/ A e. x \/ x = A ) <-> ( [. A / y ]. x e. y \/ [. A / y ]. y e. x \/ [. A / y ]. x = y ) ) ).
6:1,?: e1a 37006	|- (. A e. B ->. ( [. A / y ]. ( x e. y \/ y e. x \/ x = y ) <-> ( [. A / y ]. x e. y \/ [. A / y ]. y e. x \/ [. A / y ]. x = y ) ) ).
7:5,6: e11 37067	|- (. A e. B ->. ( [. A / y ]. ( x e. y \/ y e. x \/ x = y ) <-> ( x e. A \/ A e. x \/ x = A ) ) ).
qed:7:	|- ( A e. B -> ( [. A / y ]. ( x e. y \/ y e. x \/ x = y ) <-> ( x e. A \/ A e. x \/ x = A ) ) )

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( A e. B -> ( [. A / y ]. ( x e. y \/ y e. x \/ x = y ) <-> ( x e. A \/ A e. x \/ x = A ) ) )

Theorem

hbra2VD 37254*

Virtual deduction proof of nfra2 2771. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

1::	|- ( A. y e. B A. x e. A ph -> A. y A. y e. B A. x e. A ph )
2::	|- ( A. x e. A A. y e. B ph <-> A. y e. B A. x e. A ph )
3:1,2,?: e00 37152	|- ( A. x e. A A. y e. B ph -> A. y A. y e. B A. x e. A ph )
4:2:	|- A. y ( A. x e. A A. y e. B ph <-> A. y e. B A. x e. A ph )
5:4,?: e0a 37156	|- ( A. y A. x e. A A. y e. B ph <-> A. y A. y e. B A. x e. A ph )
qed:3,5,?: e00 37152	|- ( A. x e. A A. y e. B ph -> A. y A. x e. A A. y e. B ph )

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( A. x e. A A. y e. B ph -> A. y A. x e. A A. y e. B ph )

Theorem

tratrbVD 37255*

Virtual deduction proof of tratrb 36898. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

1::	|- (. ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) ->. ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) ).
2:1,?: e1a 37006	|- (. ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) ->. Tr A ).
3:1,?: e1a 37006	|- (. ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) ->. A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) ).
4:1,?: e1a 37006	|- (. ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) ->. B e. A ).
5::	|- (. ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) , ( x e. y /\ y e. B ) ->. ( x e. y /\ y e. B ) ).
6:5,?: e2 37010	|- (. ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) , ( x e. y /\ y e. B ) ->. x e. y ).
7:5,?: e2 37010	|- (. ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) , ( x e. y /\ y e. B ) ->. y e. B ).
8:2,7,4,?: e121 37035	|- (. ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) , ( x e. y /\ y e. B ) ->. y e. A ).
9:2,6,8,?: e122 37032	|- (. ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) , ( x e. y /\ y e. B ) ->. x e. A ).
10::	|- (. ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) , ( x e. y /\ y e. B ) , B e. x ->. B e. x ).
11:6,7,10,?: e223 37014	|- (. ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) , ( x e. y /\ y e. B ) , B e. x ->. ( x e. y /\ y e. B /\ B e. x ) ).
12:11:	|- (. ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) , ( x e. y /\ y e. B ) ->. ( B e. x -> ( x e. y /\ y e. B /\ B e. x ) ) ).
13::	|- -. ( x e. y /\ y e. B /\ B e. x )
14:12,13,?: e20 37111	|- (. ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) , ( x e. y /\ y e. B ) ->. -. B e. x ).
15::	|- (. ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) , ( x e. y /\ y e. B ) , x = B ->. x = B ).
16:7,15,?: e23 37139	|- (. ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) , ( x e. y /\ y e. B ) , x = B ->. y e. x ).
17:6,16,?: e23 37139	|- (. ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) , ( x e. y /\ y e. B ) , x = B ->. ( x e. y /\ y e. x ) ).
18:17:	|- (. ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) , ( x e. y /\ y e. B ) ->. ( x = B -> ( x e. y /\ y e. x ) ) ).
19::	|- -. ( x e. y /\ y e. x )
20:18,19,?: e20 37111	|- (. ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) , ( x e. y /\ y e. B ) ->. -. x = B ).
21:3,?: e1a 37006	|- (. ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) ->. A. y e. A A. x e. A ( x e. y \/ y e. x \/ x = y ) ).
22:21,9,4,?: e121 37035	|- (. ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) , ( x e. y /\ y e. B ) ->. [. x / x ]. [. B / y ]. ( x e. y \/ y e. x \/ x = y ) ).
23:22,?: e2 37010	|- (. ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) , ( x e. y /\ y e. B ) ->. [. B / y ]. ( x e. y \/ y e. x \/ x = y ) ).
24:4,23,?: e12 37108	|- (. ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) , ( x e. y /\ y e. B ) ->. ( x e. B \/ B e. x \/ x = B ) ).
25:14,20,24,?: e222 37015	|- (. ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) , ( x e. y /\ y e. B ) ->. x e. B ).
26:25:	|- (. ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) ->. ( ( x e. y /\ y e. B ) -> x e. B ) ).
27::	|- ( A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) -> A. y A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) )
28:27,?: e0a 37156	|- ( ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) -> A. y ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) )
29:28,26:	|- (. ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) ->. A. y ( ( x e. y /\ y e. B ) -> x e. B ) ).
30::	|- ( A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) -> A. x A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) )
31:30,?: e0a 37156	|- ( ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) -> A. x ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) )
32:31,29:	|- (. ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) ->. A. x A. y ( ( x e. y /\ y e. B ) -> x e. B ) ).
33:32,?: e1a 37006	|- (. ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) ->. Tr B ).
qed:33:	|- ( ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) -> Tr B )

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) /\ B e. A ) -> Tr B )

Theorem

al2imVD 37256

Virtual deduction proof of al2im 1690. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

1::	|- (. A. x ( ph -> ( ps -> ch ) ) ->. A. x ( ph -> ( ps -> ch ) ) ).
2:1,?: e1a 37006	|- (. A. x ( ph -> ( ps -> ch ) ) ->. ( A. x ph -> A. x ( ps -> ch ) ) ).
3::	|- ( A. x ( ps -> ch ) -> ( A. x ps -> A. x ch ) )
4:2,3,?: e10 37073	|- (. A. x ( ph -> ( ps -> ch ) ) ->. ( A. x ph -> ( A. x ps -> A. x ch ) ) ).
qed:4:	|- ( A. x ( ph -> ( ps -> ch ) ) -> ( A. x ph -> ( A. x ps -> A. x ch ) ) )

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( A. x ( ph -> ( ps -> ch ) ) -> ( A. x ph -> ( A. x ps -> A. x ch ) ) )

Theorem

syl5impVD 37257

Virtual deduction proof of syl5imp 36870. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

1::	|- (. ( ph -> ( ps -> ch ) ) ->. ( ph -> ( ps -> ch ) ) ).
2:1,?: e1a 37006	|- (. ( ph -> ( ps -> ch ) ) ->. ( ps -> ( ph -> ch ) ) ).
3::	|- (. ( ph -> ( ps -> ch ) ) ,. ( th -> ps ) ->. ( th -> ps ) ).
4:3,2,?: e21 37114	|- (. ( ph -> ( ps -> ch ) ) ,. ( th -> ps ) ->. ( th -> ( ph -> ch ) ) ).
5:4,?: e2 37010	|- (. ( ph -> ( ps -> ch ) ) ,. ( th -> ps ) ->. ( ph -> ( th -> ch ) ) ).
6:5:	|- (. ( ph -> ( ps -> ch ) ) ->. ( ( th -> ps ) -> ( ph -> ( th -> ch ) ) ) ).
qed:6:	|- ( ( ph -> ( ps -> ch ) ) -> ( ( th -> ps ) -> ( ph -> ( th -> ch ) ) ) )

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( ( ph -> ( ps -> ch ) ) -> ( ( th -> ps ) -> ( ph -> ( th -> ch ) ) ) )

Theorem

idiVD 37258

Virtual deduction proof of idiALT 36833. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

h1::	|- ph
qed:1,?: e0a 37156	|- ph

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ph   =>    |- ph

Theorem

ancomstVD 37259

Closed form of ancoms 459. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

1::	|- ( ( ph /\ ps ) <-> ( ps /\ ph ) )
qed:1,?: e0a 37156	|- ( ( ( ph /\ ps ) -> ch ) <-> ( ( ps /\ ph ) -> ch ) )

The proof of ancomst 458 is derived automatically from it. (Contributed by Alan Sare, 25-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( ( ( ph /\ ps ) -> ch ) <-> ( ( ps /\ ph ) -> ch ) )

Theorem

ssralv2VD 37260*

Quantification restricted to a subclass for two quantifiers. ssralv 3461 for two quantifiers. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. ssralv2 36889 is ssralv2VD 37260 without virtual deductions and was automatically derived from ssralv2VD 37260.

1::	|- (. ( A C_ B /\ C C_ D ) ->. ( A C_ B /\ C C_ D ) ).
2::	|- (. ( A C_ B /\ C C_ D ) ,. A. x e. B A. y e. D ph ->. A. x e. B A. y e. D ph ).
3:1:	|- (. ( A C_ B /\ C C_ D ) ->. A C_ B ).
4:3,2:	|- (. ( A C_ B /\ C C_ D ) ,. A. x e. B A. y e. D ph ->. A. x e. A A. y e. D ph ).
5:4:	|- (. ( A C_ B /\ C C_ D ) ,. A. x e. B A. y e. D ph ->. A. x ( x e. A -> A. y e. D ph ) ).
6:5:	|- (. ( A C_ B /\ C C_ D ) ,. A. x e. B A. y e. D ph ->. ( x e. A -> A. y e. D ph ) ).
7::	|- (. ( A C_ B /\ C C_ D ) ,. A. x e. B A. y e. D ph , x e. A ->. x e. A ).
8:7,6:	|- (. ( A C_ B /\ C C_ D ) ,. A. x e. B A. y e. D ph , x e. A ->. A. y e. D ph ).
9:1:	|- (. ( A C_ B /\ C C_ D ) ->. C C_ D ).
10:9,8:	|- (. ( A C_ B /\ C C_ D ) ,. A. x e. B A. y e. D ph , x e. A ->. A. y e. C ph ).
11:10:	|- (. ( A C_ B /\ C C_ D ) ,. A. x e. B A. y e. D ph ->. ( x e. A -> A. y e. C ph ) ).
12::	|- ( ( A C_ B /\ C C_ D ) -> A. x ( A C_ B /\ C C_ D ) )
13::	|- ( A. x e. B A. y e. D ph -> A. x A. x e. B A. y e. D ph )
14:12,13,11:	|- (. ( A C_ B /\ C C_ D ) ,. A. x e. B A. y e. D ph ->. A. x ( x e. A -> A. y e. C ph ) ).
15:14:	|- (. ( A C_ B /\ C C_ D ) ,. A. x e. B A. y e. D ph ->. A. x e. A A. y e. C ph ).
16:15:	|- (. ( A C_ B /\ C C_ D ) ->. ( A. x e. B A. y e. D ph -> A. x e. A A. y e. C ph ) ).
qed:16:	|- ( ( A C_ B /\ C C_ D ) -> ( A. x e. B A. y e. D ph -> A. x e. A A. y e. C ph ) )

(Contributed by Alan Sare, 10-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( ( A C_ B /\ C C_ D ) -> ( A. x e. B A. y e. D ph -> A. x e. A A. y e. C ph ) )

Theorem

ordelordALTVD 37261

An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 5424 using the Axiom of Regularity indirectly through dford2 8112. dford2 is a weaker definition of ordinal number. Given the Axiom of Regularity, it need not be assumed that _E Fr A because this is inferred by the Axiom of Regularity. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. ordelordALT 36899 is ordelordALTVD 37261 without virtual deductions and was automatically derived from ordelordALTVD 37261 using the tools program translate..without..overwriting.cmd and Metamath's minimize command.

1::	|- (. ( Ord A /\ B e. A ) ->. ( Ord A /\ B e. A ) ).
2:1:	|- (. ( Ord A /\ B e. A ) ->. Ord A ).
3:1:	|- (. ( Ord A /\ B e. A ) ->. B e. A ).
4:2:	|- (. ( Ord A /\ B e. A ) ->. Tr A ).
5:2:	|- (. ( Ord A /\ B e. A ) ->. A. x e. A A. y e. A ( x e. y \/ x = y \/ y e. x ) ).
6:4,3:	|- (. ( Ord A /\ B e. A ) ->. B C_ A ).
7:6,6,5:	|- (. ( Ord A /\ B e. A ) ->. A. x e. B A. y e. B ( x e. y \/ x = y \/ y e. x ) ).
8::	|- ( ( x e. y \/ x = y \/ y e. x ) <-> ( x e. y \/ y e. x \/ x = y ) )
9:8:	|- A. y ( ( x e. y \/ x = y \/ y e. x ) <-> ( x e. y \/ y e. x \/ x = y ) )
10:9:	|- A. y e. A ( ( x e. y \/ x = y \/ y e. x ) <-> ( x e. y \/ y e. x \/ x = y ) )
11:10:	|- ( A. y e. A ( x e. y \/ x = y \/ y e. x ) <-> A. y e. A ( x e. y \/ y e. x \/ x = y ) )
12:11:	|- A. x ( A. y e. A ( x e. y \/ x = y \/ y e. x ) <-> A. y e. A ( x e. y \/ y e. x \/ x = y ) )
13:12:	|- A. x e. A ( A. y e. A ( x e. y \/ x = y \/ y e. x ) <-> A. y e. A ( x e. y \/ y e. x \/ x = y ) )
14:13:	|- ( A. x e. A A. y e. A ( x e. y \/ x = y \/ y e. x ) <-> A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) )
15:14,5:	|- (. ( Ord A /\ B e. A ) ->. A. x e. A A. y e. A ( x e. y \/ y e. x \/ x = y ) ).
16:4,15,3:	|- (. ( Ord A /\ B e. A ) ->. Tr B ).
17:16,7:	|- (. ( Ord A /\ B e. A ) ->. Ord B ).
qed:17:	|- ( ( Ord A /\ B e. A ) -> Ord B )

(Contributed by Alan Sare, 12-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( ( Ord A /\ B e. A ) -> Ord B )

Theorem

equncomVD 37262

If a class equals the union of two other classes, then it equals the union of those two classes commuted. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. equncom 3547 is equncomVD 37262 without virtual deductions and was automatically derived from equncomVD 37262.

1::	|- (. A = ( B u. C ) ->. A = ( B u. C ) ).
2::	|- ( B u. C ) = ( C u. B )
3:1,2:	|- (. A = ( B u. C ) ->. A = ( C u. B ) ).
4:3:	|- ( A = ( B u. C ) -> A = ( C u. B ) )
5::	|- (. A = ( C u. B ) ->. A = ( C u. B ) ).
6:5,2:	|- (. A = ( C u. B ) ->. A = ( B u. C ) ).
7:6:	|- ( A = ( C u. B ) -> A = ( B u. C ) )
8:4,7:	|- ( A = ( B u. C ) <-> A = ( C u. B ) )

(Contributed by Alan Sare, 17-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( A = ( B u. C ) <-> A = ( C u. B ) )

Theorem

equncomiVD 37263

Inference form of equncom 3547. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. equncomi 3548 is equncomiVD 37263 without virtual deductions and was automatically derived from equncomiVD 37263.

h1::	|- A = ( B u. C )
qed:1:	|- A = ( C u. B )

(Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

|- A = ( B u. C )   =>    |- A = ( C u. B )

Theorem

sucidALTVD 37264

A set belongs to its successor. Alternate proof of sucid 5481. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sucidALT 37265 is sucidALTVD 37264 without virtual deductions and was automatically derived from sucidALTVD 37264. This proof illustrates that completeusersproof.cmd will generate a Metamath proof from any User's Proof which is "conventional" in the sense that no step is a virtual deduction, provided that all necessary unification theorems and transformation deductions are in set.mm. completeusersproof.cmd automatically converts such a conventional proof into a Virtual Deduction proof for which each step happens to be a 0-virtual hypothesis virtual deduction. The user does not need to search for reference theorem labels or deduction labels nor does he(she) need to use theorems and deductions which unify with reference theorems and deductions in set.mm. All that is necessary is that each theorem or deduction of the User's Proof unifies with some reference theorem or deduction in set.mm or is a semantic variation of some theorem or deduction which unifies with some reference theorem or deduction in set.mm. The definition of "semantic variation" has not been precisely defined. If it is obvious that a theorem or deduction has the same meaning as another theorem or deduction, then it is a semantic variation of the latter theorem or deduction. For example, step 4 of the User's Proof is a semantic variation of the definition (axiom) suc A = ( A u. { A } ), which unifies with df-suc 5408, a reference definition (axiom) in set.mm. Also, a theorem or deduction is said to be a semantic variation of another theorem or deduction if it is obvious upon cursory inspection that it has the same meaning as a weaker form of the latter theorem or deduction. For example, the deduction Ord A infers A. x e. A A. y e. A ( x e. y \/ x = y \/ y e. x ) is a semantic variation of the theorem ( Ord A <-> ( Tr A /\ A. x e. A A. y e. A ( x e. y \/ x = y \/ y e. x ) ) ), which unifies with the set.mm reference definition (axiom) dford2 8112.

h1::	|- A e. _V
2:1:	|- A e. { A }
3:2:	|- A e. ( { A } u. A )
4::	|- suc A = ( { A } u. A )
qed:3,4:	|- A e. suc A

(Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

|- A e. _V   =>    |- A e. suc A

Theorem

sucidALT 37265

A set belongs to its successor. This proof was automatically derived from sucidALTVD 37264 using translate_without_overwriting.cmd and minimizing. (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

|- A e. _V   =>    |- A e. suc A

Theorem

sucidVD 37266

A set belongs to its successor. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sucid 5481 is sucidVD 37266 without virtual deductions and was automatically derived from sucidVD 37266.

h1::	|- A e. _V
2:1:	|- A e. { A }
3:2:	|- A e. ( A u. { A } )
4::	|- suc A = ( A u. { A } )
qed:3,4:	|- A e. suc A

(Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

|- A e. _V   =>    |- A e. suc A

Theorem

imbi12VD 37267

Implication form of imbi12i 332. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. imbi12 328 is imbi12VD 37267 without virtual deductions and was automatically derived from imbi12VD 37267.

1::	|- (. ( ph <-> ps ) ->. ( ph <-> ps ) ).
2::	|- (. ( ph <-> ps ) ,. ( ch <-> th ) ->. ( ch <-> th ) ).
3::	|- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ph -> ch ) ->. ( ph -> ch ) ).
4:1,3:	|- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ph -> ch ) ->. ( ps -> ch ) ).
5:2,4:	|- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ph -> ch ) ->. ( ps -> th ) ).
6:5:	|- (. ( ph <-> ps ) ,. ( ch <-> th ) ->. ( ( ph -> ch ) -> ( ps -> th ) ) ).
7::	|- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ps -> th ) ->. ( ps -> th ) ).
8:1,7:	|- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ps -> th ) ->. ( ph -> th ) ).
9:2,8:	|- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ps -> th ) ->. ( ph -> ch ) ).
10:9:	|- (. ( ph <-> ps ) ,. ( ch <-> th ) ->. ( ( ps -> th ) -> ( ph -> ch ) ) ).
11:6,10:	|- (. ( ph <-> ps ) ,. ( ch <-> th ) ->. ( ( ph -> ch ) <-> ( ps -> th ) ) ).
12:11:	|- (. ( ph <-> ps ) ->. ( ( ch <-> th ) -> ( ( ph -> ch ) <-> ( ps -> th ) ) ) ).
qed:12:	|- ( ( ph <-> ps ) -> ( ( ch <-> th ) -> ( ( ph -> ch ) <-> ( ps -> th ) ) ) )

(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( ( ph <-> ps ) -> ( ( ch <-> th ) -> ( ( ph -> ch ) <-> ( ps -> th ) ) ) )

Theorem

imbi13VD 37268

Join three logical equivalences to form equivalence of implications. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. imbi13 36878 is imbi13VD 37268 without virtual deductions and was automatically derived from imbi13VD 37268.

1::	|- (. ( ph <-> ps ) ->. ( ph <-> ps ) ).
2::	|- (. ( ph <-> ps ) ,. ( ch <-> th ) ->. ( ch <-> th ) ).
3::	|- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ta <-> et ) ->. ( ta <-> et ) ).
4:2,3:	|- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ta <-> et ) ->. ( ( ch -> ta ) <-> ( th -> et ) ) ).
5:1,4:	|- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ta <-> et ) ->. ( ( ph -> ( ch -> ta ) ) <-> ( ps -> ( th -> et ) ) ) ).
6:5:	|- (. ( ph <-> ps ) ,. ( ch <-> th ) ->. ( ( ta <-> et ) -> ( ( ph -> ( ch -> ta ) ) <-> ( ps -> ( th -> et ) ) ) ) ).
7:6:	|- (. ( ph <-> ps ) ->. ( ( ch <-> th ) -> ( ( ta <-> et ) -> ( ( ph -> ( ch -> ta ) ) <-> ( ps -> ( th -> et ) ) ) ) ) ).
qed:7:	|- ( ( ph <-> ps ) -> ( ( ch <-> th ) -> ( ( ta <-> et ) -> ( ( ph -> ( ch -> ta ) ) <-> ( ps -> ( th -> et ) ) ) ) ) )

(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( ( ph <-> ps ) -> ( ( ch <-> th ) -> ( ( ta <-> et ) -> ( ( ph -> ( ch -> ta ) ) <-> ( ps -> ( th -> et ) ) ) ) ) )

Theorem

sbcim2gVD 37269

Distribution of class substitution over a left-nested implication. Similar to sbcimg 3277. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sbcim2g 36900 is sbcim2gVD 37269 without virtual deductions and was automatically derived from sbcim2gVD 37269.

1::	|- (. A e. B ->. A e. B ).
2::	|- (. A e. B ,. [. A / x ]. ( ph -> ( ps -> ch ) ) ->. [. A / x ]. ( ph -> ( ps -> ch ) ) ).
3:1,2:	|- (. A e. B ,. [. A / x ]. ( ph -> ( ps -> ch ) ) ->. ( [. A / x ]. ph -> [. A / x ]. ( ps -> ch ) ) ).
4:1:	|- (. A e. B ->. ( [. A / x ]. ( ps -> ch ) <-> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ).
5:3,4:	|- (. A e. B ,. [. A / x ]. ( ph -> ( ps -> ch ) ) ->. ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ).
6:5:	|- (. A e. B ->. ( [. A / x ]. ( ph -> ( ps -> ch ) ) -> ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ) ).
7::	|- (. A e. B ,. ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ->. ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ).
8:4,7:	|- (. A e. B ,. ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ->. ( [. A / x ]. ph -> [. A / x ]. ( ps -> ch ) ) ).
9:1:	|- (. A e. B ->. ( [. A / x ]. ( ph -> ( ps -> ch ) ) <-> ( [. A / x ]. ph -> [. A / x ]. ( ps -> ch ) ) ) ).
10:8,9:	|- (. A e. B ,. ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ->. [. A / x ]. ( ph -> ( ps -> ch ) ) ).
11:10:	|- (. A e. B ->. ( ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) -> [. A / x ]. ( ph -> ( ps -> ch ) ) ) ).
12:6,11:	|- (. A e. B ->. ( [. A / x ]. ( ph -> ( ps -> ch ) ) <-> ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ) ).
qed:12:	|- ( A e. B -> ( [. A / x ]. ( ph -> ( ps -> ch ) ) <-> ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ) )

(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( A e. B -> ( [. A / x ]. ( ph -> ( ps -> ch ) ) <-> ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ) )

Theorem

sbcbiVD 37270

Implication form of sbcbiiOLD 36893. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sbcbi 36901 is sbcbiVD 37270 without virtual deductions and was automatically derived from sbcbiVD 37270.

1::	|- (. A e. B ->. A e. B ).
2::	|- (. A e. B ,. A. x ( ph <-> ps ) ->. A. x ( ph <-> ps ) ).
3:1,2:	|- (. A e. B ,. A. x ( ph <-> ps ) ->. [. A / x ]. ( ph <-> ps ) ).
4:1,3:	|- (. A e. B ,. A. x ( ph <-> ps ) ->. ( [. A / x ]. ph <-> [. A / x ]. ps ) ).
5:4:	|- (. A e. B ->. ( A. x ( ph <-> ps ) -> ( [. A / x ]. ph <-> [. A / x ]. ps ) ) ).
qed:5:	|- ( A e. B -> ( A. x ( ph <-> ps ) -> ( [. A / x ]. ph <-> [. A / x ]. ps ) ) )

(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( A e. B -> ( A. x ( ph <-> ps ) -> ( [. A / x ]. ph <-> [. A / x ]. ps ) ) )

Theorem

trsbcVD 37271*

Formula-building inference rule for class substitution, substituting a class variable for the setvar variable of the transitivity predicate. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. trsbc 36902 is trsbcVD 37271 without virtual deductions and was automatically derived from trsbcVD 37271.

1::	|- (. A e. B ->. A e. B ).
2:1:	|- (. A e. B ->. ( [. A / x ]. z e. y <-> z e. y ) ).
3:1:	|- (. A e. B ->. ( [. A / x ]. y e. x <-> y e. A ) ).
4:1:	|- (. A e. B ->. ( [. A / x ]. z e. x <-> z e. A ) ).
5:1,2,3,4:	|- (. A e. B ->. ( ( [. A / x ]. z e. y -> ( [. A / x ]. y e. x -> [. A / x ]. z e. x ) ) <-> ( z e. y -> ( y e. A -> z e. A ) ) ) ).
6:1:	|- (. A e. B ->. ( [. A / x ]. ( z e. y -> ( y e. x -> z e. x ) ) <-> ( [. A / x ]. z e. y -> ( [. A / x ]. y e. x -> [. A / x ]. z e. x ) ) ) ).
7:5,6:	|- (. A e. B ->. ( [. A / x ]. ( z e. y -> ( y e. x -> z e. x ) ) <-> ( z e. y -> ( y e. A -> z e. A ) ) ) ).
8::	|- ( ( z e. y -> ( y e. A -> z e. A ) ) <-> ( ( z e. y /\ y e. A ) -> z e. A ) )
9:7,8:	|- (. A e. B ->. ( [. A / x ]. ( z e. y -> ( y e. x -> z e. x ) ) <-> ( ( z e. y /\ y e. A ) -> z e. A ) ) ).
10::	|- ( ( z e. y -> ( y e. x -> z e. x ) ) <-> ( ( z e. y /\ y e. x ) -> z e. x ) )
11:10:	|- A. x ( ( z e. y -> ( y e. x -> z e. x ) ) <-> ( ( z e. y /\ y e. x ) -> z e. x ) )
12:1,11:	|- (. A e. B ->. ( [. A / x ]. ( z e. y -> ( y e. x -> z e. x ) ) <-> [. A / x ]. ( ( z e. y /\ y e. x ) -> z e. x ) ) ).
13:9,12:	|- (. A e. B ->. ( [. A / x ]. ( ( z e. y /\ y e. x ) -> z e. x ) <-> ( ( z e. y /\ y e. A ) -> z e. A ) ) ).
14:13:	|- (. A e. B ->. A. y ( [. A / x ]. ( ( z e. y /\ y e. x ) -> z e. x ) <-> ( ( z e. y /\ y e. A ) -> z e. A ) ) ).
15:14:	|- (. A e. B ->. ( A. y [. A / x ]. ( ( z e. y /\ y e. x ) -> z e. x ) <-> A. y ( ( z e. y /\ y e. A ) -> z e. A ) ) ).
16:1:	|- (. A e. B ->. ( [. A / x ]. A. y ( ( z e. y /\ y e. x ) -> z e. x ) <-> A. y [. A / x ]. ( ( z e. y /\ y e. x ) -> z e. x ) ) ).
17:15,16:	|- (. A e. B ->. ( [. A / x ]. A. y ( ( z e. y /\ y e. x ) -> z e. x ) <-> A. y ( ( z e. y /\ y e. A ) -> z e. A ) ) ).
18:17:	|- (. A e. B ->. A. z ( [. A / x ]. A. y ( ( z e. y /\ y e. x ) -> z e. x ) <-> A. y ( ( z e. y /\ y e. A ) -> z e. A ) ) ).
19:18:	|- (. A e. B ->. ( A. z [. A / x ]. A. y ( ( z e. y /\ y e. x ) -> z e. x ) <-> A. z A. y ( ( z e. y /\ y e. A ) -> z e. A ) ) ).
20:1:	|- (. A e. B ->. ( [. A / x ]. A. z A. y ( ( z e. y /\ y e. x ) -> z e. x ) <-> A. z [. A / x ]. A. y ( ( z e. y /\ y e. x ) -> z e. x ) ) ).
21:19,20:	|- (. A e. B ->. ( [. A / x ]. A. z A. y ( ( z e. y /\ y e. x ) -> z e. x ) <-> A. z A. y ( ( z e. y /\ y e. A ) -> z e. A ) ) ).
22::	|- ( Tr A <-> A. z A. y ( ( z e. y /\ y e. A ) -> z e. A ) )
23:21,22:	|- (. A e. B ->. ( [. A / x ]. A. z A. y ( ( z e. y /\ y e. x ) -> z e. x ) <-> Tr A ) ).
24::	|- ( Tr x <-> A. z A. y ( ( z e. y /\ y e. x ) -> z e. x ) )
25:24:	|- A. x ( Tr x <-> A. z A. y ( ( z e. y /\ y e. x ) -> z e. x ) )
26:1,25:	|- (. A e. B ->. ( [. A / x ]. Tr x <-> [. A / x ]. A. z A. y ( ( z e. y /\ y e. x ) -> z e. x ) ) ).
27:23,26:	|- (. A e. B ->. ( [. A / x ]. Tr x <-> Tr A ) ).
qed:27:	|- ( A e. B -> ( [. A / x ]. Tr x <-> Tr A ) )

(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( A e. B -> ( [. A / x ]. Tr x <-> Tr A ) )

Theorem

truniALTVD 37272*

The union of a class of transitive sets is transitive. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. truniALT 36903 is truniALTVD 37272 without virtual deductions and was automatically derived from truniALTVD 37272.

1::	|- (. A. x e. A Tr x ->. A. x e. A Tr x ).
2::	|- (. A. x e. A Tr x ,. ( z e. y /\ y e. U. A ) ->. ( z e. y /\ y e. U. A ) ).
3:2:	|- (. A. x e. A Tr x ,. ( z e. y /\ y e. U. A ) ->. z e. y ).
4:2:	|- (. A. x e. A Tr x ,. ( z e. y /\ y e. U. A ) ->. y e. U. A ).
5:4:	|- (. A. x e. A Tr x ,. ( z e. y /\ y e. U. A ) ->. E. q ( y e. q /\ q e. A ) ).
6::	|- (. A. x e. A Tr x ,. ( z e. y /\ y e. U. A ) , ( y e. q /\ q e. A ) ->. ( y e. q /\ q e. A ) ).
7:6:	|- (. A. x e. A Tr x ,. ( z e. y /\ y e. U. A ) , ( y e. q /\ q e. A ) ->. y e. q ).
8:6:	|- (. A. x e. A Tr x ,. ( z e. y /\ y e. U. A ) , ( y e. q /\ q e. A ) ->. q e. A ).
9:1,8:	|- (. A. x e. A Tr x ,. ( z e. y /\ y e. U. A ) , ( y e. q /\ q e. A ) ->. [ q / x ] Tr x ).
10:8,9:	|- (. A. x e. A Tr x ,. ( z e. y /\ y e. U. A ) , ( y e. q /\ q e. A ) ->. Tr q ).
11:3,7,10:	|- (. A. x e. A Tr x ,. ( z e. y /\ y e. U. A ) , ( y e. q /\ q e. A ) ->. z e. q ).
12:11,8:	|- (. A. x e. A Tr x ,. ( z e. y /\ y e. U. A ) , ( y e. q /\ q e. A ) ->. z e. U. A ).
13:12:	|- (. A. x e. A Tr x ,. ( z e. y /\ y e. U. A ) ->. ( ( y e. q /\ q e. A ) -> z e. U. A ) ).
14:13:	|- (. A. x e. A Tr x ,. ( z e. y /\ y e. U. A ) ->. A. q ( ( y e. q /\ q e. A ) -> z e. U. A ) ).
15:14:	|- (. A. x e. A Tr x ,. ( z e. y /\ y e. U. A ) ->. ( E. q ( y e. q /\ q e. A ) -> z e. U. A ) ).
16:5,15:	|- (. A. x e. A Tr x ,. ( z e. y /\ y e. U. A ) ->. z e. U. A ).
17:16:	|- (. A. x e. A Tr x ->. ( ( z e. y /\ y e. U. A ) -> z e. U. A ) ).
18:17:	|- (. A. x e. A Tr x ->. A. z A. y ( ( z e. y /\ y e. U. A ) -> z e. U. A ) ).
19:18:	|- (. A. x e. A Tr x ->. Tr U. A ).
qed:19:	|- ( A. x e. A Tr x -> Tr U. A )

(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( A. x e. A Tr x -> Tr U. A )

Theorem

ee33VD 37273

Non-virtual deduction form of e33 37118. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. ee33 36879 is ee33VD 37273 without virtual deductions and was automatically derived from ee33VD 37273.

h1::	|- ( ph -> ( ps -> ( ch -> th ) ) )
h2::	|- ( ph -> ( ps -> ( ch -> ta ) ) )
h3::	|- ( th -> ( ta -> et ) )
4:1,3:	|- ( ph -> ( ps -> ( ch -> ( ta -> et ) ) ) )
5:4:	|- ( ta -> ( ph -> ( ps -> ( ch -> et ) ) ) )
6:2,5:	|- ( ph -> ( ps -> ( ch -> ( ph -> ( ps -> ( ch -> et ) ) ) ) ) )
7:6:	|- ( ps -> ( ch -> ( ph -> ( ps -> ( ch -> et ) ) ) ) )
8:7:	|- ( ch -> ( ph -> ( ps -> ( ch -> et ) ) ) )
qed:8:	|- ( ph -> ( ps -> ( ch -> et ) ) )

(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( ph -> ( ps -> ( ch -> th ) ) )   &    |- ( ph -> ( ps -> ( ch -> ta ) ) )   &    |- ( th -> ( ta -> et ) )   =>    |- ( ph -> ( ps -> ( ch -> et ) ) )

Theorem

trintALTVD 37274*

The intersection of a class of transitive sets is transitive. Virtual deduction proof of trintALT 37275. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. trintALT 37275 is trintALTVD 37274 without virtual deductions and was automatically derived from trintALTVD 37274.

1::	|- (. A. x e. A Tr x ->. A. x e. A Tr x ).
2::	|- (. A. x e. A Tr x ,. ( z e. y /\ y e. |^| A ) ->. ( z e. y /\ y e. |^| A ) ).
3:2:	|- (. A. x e. A Tr x ,. ( z e. y /\ y e. |^| A ) ->. z e. y ).
4:2:	|- (. A. x e. A Tr x ,. ( z e. y /\ y e. |^| A ) ->. y e. |^| A ).
5:4:	|- (. A. x e. A Tr x ,. ( z e. y /\ y e. |^| A ) ->. A. q e. A y e. q ).
6:5:	|- (. A. x e. A Tr x ,. ( z e. y /\ y e. |^| A ) ->. ( q e. A -> y e. q ) ).
7::	|- (. A. x e. A Tr x ,. ( z e. y /\ y e. |^| A ) , q e. A ->. q e. A ).
8:7,6:	|- (. A. x e. A Tr x ,. ( z e. y /\ y e. |^| A ) , q e. A ->. y e. q ).
9:7,1:	|- (. A. x e. A Tr x ,. ( z e. y /\ y e. |^| A ) , q e. A ->. [ q / x ] Tr x ).
10:7,9:	|- (. A. x e. A Tr x ,. ( z e. y /\ y e. |^| A ) , q e. A ->. Tr q ).
11:10,3,8:	|- (. A. x e. A Tr x ,. ( z e. y /\ y e. |^| A ) , q e. A ->. z e. q ).
12:11:	|- (. A. x e. A Tr x ,. ( z e. y /\ y e. |^| A ) ->. ( q e. A -> z e. q ) ).
13:12:	|- (. A. x e. A Tr x ,. ( z e. y /\ y e. |^| A ) ->. A. q ( q e. A -> z e. q ) ).
14:13:	|- (. A. x e. A Tr x ,. ( z e. y /\ y e. |^| A ) ->. A. q e. A z e. q ).
15:3,14:	|- (. A. x e. A Tr x ,. ( z e. y /\ y e. |^| A ) ->. z e. |^| A ).
16:15:	|- (. A. x e. A Tr x ->. ( ( z e. y /\ y e. |^| A ) -> z e. |^| A ) ).
17:16:	|- (. A. x e. A Tr x ->. A. z A. y ( ( z e. y /\ y e. |^| A ) -> z e. |^| A ) ).
18:17:	|- (. A. x e. A Tr x ->. Tr |^| A ).
qed:18:	|- ( A. x e. A Tr x -> Tr |^| A )

(Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( A. x e. A Tr x -> Tr |^| A )

Theorem

trintALT 37275*

The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. trintALT 37275 is an alternate proof of trint 4484. trintALT 37275 is trintALTVD 37274 without virtual deductions and was automatically derived from trintALTVD 37274 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( A. x e. A Tr x -> Tr |^| A )

Theorem

undif3VD 37276

The first equality of Exercise 13 of [TakeutiZaring] p. 22. Virtual deduction proof of undif3 3672. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. undif3 3672 is undif3VD 37276 without virtual deductions and was automatically derived from undif3VD 37276.

1::	|- ( x e. ( A u. ( B \ C ) ) <-> ( x e. A \/ x e. ( B \ C ) ) )
2::	|- ( x e. ( B \ C ) <-> ( x e. B /\ -. x e. C ) )
3:2:	|- ( ( x e. A \/ x e. ( B \ C ) ) <-> ( x e. A \/ ( x e. B /\ -. x e. C ) ) )
4:1,3:	|- ( x e. ( A u. ( B \ C ) ) <-> ( x e. A \/ ( x e. B /\ -. x e. C ) ) )
5::	|- (. x e. A ->. x e. A ).
6:5:	|- (. x e. A ->. ( x e. A \/ x e. B ) ).
7:5:	|- (. x e. A ->. ( -. x e. C \/ x e. A ) ).
8:6,7:	|- (. x e. A ->. ( ( x e. A \/ x e. B ) /\ ( -. x e. C \/ x e. A ) ) ).
9:8:	|- ( x e. A -> ( ( x e. A \/ x e. B ) /\ ( -. x e. C \/ x e. A ) ) )
10::	|- (. ( x e. B /\ -. x e. C ) ->. ( x e. B /\ -. x e. C ) ).
11:10:	|- (. ( x e. B /\ -. x e. C ) ->. x e. B ).
12:10:	|- (. ( x e. B /\ -. x e. C ) ->. -. x e. C ).
13:11:	|- (. ( x e. B /\ -. x e. C ) ->. ( x e. A \/ x e. B ) ).
14:12:	|- (. ( x e. B /\ -. x e. C ) ->. ( -. x e. C \/ x e. A ) ).
15:13,14:	|- (. ( x e. B /\ -. x e. C ) ->. ( ( x e. A \/ x e. B ) /\ ( -. x e. C \/ x e. A ) ) ).
16:15:	|- ( ( x e. B /\ -. x e. C ) -> ( ( x e. A \/ x e. B ) /\ ( -. x e. C \/ x e. A ) ) )
17:9,16:	|- ( ( x e. A \/ ( x e. B /\ -. x e. C ) ) -> ( ( x e. A \/ x e. B ) /\ ( -. x e. C \/ x e. A ) ) )
18::	|- (. ( x e. A /\ -. x e. C ) ->. ( x e. A /\ -. x e. C ) ).
19:18:	|- (. ( x e. A /\ -. x e. C ) ->. x e. A ).
20:18:	|- (. ( x e. A /\ -. x e. C ) ->. -. x e. C ).
21:18:	|- (. ( x e. A /\ -. x e. C ) ->. ( x e. A \/ ( x e. B /\ -. x e. C ) ) ).
22:21:	|- ( ( x e. A /\ -. x e. C ) -> ( x e. A \/ ( x e. B /\ -. x e. C ) ) )
23::	|- (. ( x e. A /\ x e. A ) ->. ( x e. A /\ x e. A ) ).
24:23:	|- (. ( x e. A /\ x e. A ) ->. x e. A ).
25:24:	|- (. ( x e. A /\ x e. A ) ->. ( x e. A \/ ( x e. B /\ -. x e. C ) ) ).
26:25:	|- ( ( x e. A /\ x e. A ) -> ( x e. A \/ ( x e. B /\ -. x e. C ) ) )
27:10:	|- (. ( x e. B /\ -. x e. C ) ->. ( x e. A \/ ( x e. B /\ -. x e. C ) ) ).
28:27:	|- ( ( x e. B /\ -. x e. C ) -> ( x e. A \/ ( x e. B /\ -. x e. C ) ) )
29::	|- (. ( x e. B /\ x e. A ) ->. ( x e. B /\ x e. A ) ).
30:29:	|- (. ( x e. B /\ x e. A ) ->. x e. A ).
31:30:	|- (. ( x e. B /\ x e. A ) ->. ( x e. A \/ ( x e. B /\ -. x e. C ) ) ).
32:31:	|- ( ( x e. B /\ x e. A ) -> ( x e. A \/ ( x e. B /\ -. x e. C ) ) )
33:22,26:	|- ( ( ( x e. A /\ -. x e. C ) \/ ( x e. A /\ x e. A ) ) -> ( x e. A \/ ( x e. B /\ -. x e. C ) ) )
34:28,32:	|- ( ( ( x e. B /\ -. x e. C ) \/ ( x e. B /\ x e. A ) ) -> ( x e. A \/ ( x e. B /\ -. x e. C ) ) )
35:33,34:	|- ( ( ( ( x e. A /\ -. x e. C ) \/ ( x e. A /\ x e. A ) ) \/ ( ( x e. B /\ -. x e. C ) \/ ( x e. B /\ x e. A ) ) ) -> ( x e. A \/ ( x e. B /\ -. x e. C ) ) )
36::	|- ( ( ( ( x e. A /\ -. x e. C ) \/ ( x e. A /\ x e. A ) ) \/ ( ( x e. B /\ -. x e. C ) \/ ( x e. B /\ x e. A ) ) ) <-> ( ( x e. A \/ x e. B ) /\ ( -. x e. C \/ x e. A ) ) )
37:36,35:	|- ( ( ( x e. A \/ x e. B ) /\ ( -. x e. C \/ x e. A ) ) -> ( x e. A \/ ( x e. B /\ -. x e. C ) ) )
38:17,37:	|- ( ( x e. A \/ ( x e. B /\ -. x e. C ) ) <-> ( ( x e. A \/ x e. B ) /\ ( -. x e. C \/ x e. A ) ) )
39::	|- ( x e. ( C \ A ) <-> ( x e. C /\ -. x e. A ) )
40:39:	|- ( -. x e. ( C \ A ) <-> -. ( x e. C /\ -. x e. A ) )
41::	|- ( -. ( x e. C /\ -. x e. A ) <-> ( -. x e. C \/ x e. A ) )
42:40,41:	|- ( -. x e. ( C \ A ) <-> ( -. x e. C \/ x e. A ) )
43::	|- ( x e. ( A u. B ) <-> ( x e. A \/ x e. B ) )
44:43,42:	|- ( ( x e. ( A u. B ) /\ -. x e. ( C \ A ) ) <-> ( ( x e. A \/ x e. B ) /\ ( -. x e. C /\ x e. A ) ) )
45::	|- ( x e. ( ( A u. B ) \ ( C \ A ) ) <-> ( x e. ( A u. B ) /\ -. x e. ( C \ A ) ) )
46:45,44:	|- ( x e. ( ( A u. B ) \ ( C \ A ) ) <-> ( ( x e. A \/ x e. B ) /\ ( -. x e. C \/ x e. A ) ) )
47:4,38:	|- ( x e. ( A u. ( B \ C ) ) <-> ( ( x e. A \/ x e. B ) /\ ( -. x e. C \/ x e. A ) ) )
48:46,47:	|- ( x e. ( A u. ( B \ C ) ) <-> x e. ( ( A u. B ) \ ( C \ A ) ) )
49:48:	|- A. x ( x e. ( A u. ( B \ C ) ) <-> x e. ( ( A u. B ) \ ( C \ A ) ) )
qed:49:	|- ( A u. ( B \ C ) ) = ( ( A u. B ) \ ( C \ A ) )

(Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( A u. ( B \ C ) ) = ( ( A u. B ) \ ( C \ A ) )

Theorem

sbcssgVD 37277

Virtual deduction proof of sbcssg 3848. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sbcssg 3848 is sbcssgVD 37277 without virtual deductions and was automatically derived from sbcssgVD 37277.

1::	|- (. A e. B ->. A e. B ).
2:1:	|- (. A e. B ->. ( [. A / x ]. y e. C <-> y e. [_ A / x ]_ C ) ).
3:1:	|- (. A e. B ->. ( [. A / x ]. y e. D <-> y e. [_ A / x ]_ D ) ).
4:2,3:	|- (. A e. B ->. ( ( [. A / x ]. y e. C -> [. A / x ]. y e. D ) <-> ( y e. [_ A / x ]_ C -> y e. [_ A / x ]_ D ) ) ).
5:1:	|- (. A e. B ->. ( [. A / x ]. ( y e. C -> y e. D ) <-> ( [. A / x ]. y e. C -> [. A / x ]. y e. D ) ) ).
6:4,5:	|- (. A e. B ->. ( [. A / x ]. ( y e. C -> y e. D ) <-> ( y e. [_ A / x ]_ C -> y e. [_ A / x ]_ D ) ) ).
7:6:	|- (. A e. B ->. A. y ( [. A / x ]. ( y e. C -> y e. D ) <-> ( y e. [_ A / x ]_ C -> y e. [_ A / x ]_ D ) ) ).
8:7:	|- (. A e. B ->. ( A. y [. A / x ]. ( y e. C -> y e. D ) <-> A. y ( y e. [_ A / x ]_ C -> y e. [_ A / x ]_ D ) ) ).
9:1:	|- (. A e. B ->. ( [. A / x ]. A. y ( y e. C -> y e. D ) <-> A. y [. A / x ]. ( y e. C -> y e. D ) ) ).
10:8,9:	|- (. A e. B ->. ( [. A / x ]. A. y ( y e. C -> y e. D ) <-> A. y ( y e. [_ A / x ]_ C -> y e. [_ A / x ]_ D ) ) ).
11::	|- ( C C_ D <-> A. y ( y e. C -> y e. D ) )
110:11:	|- A. x ( C C_ D <-> A. y ( y e. C -> y e. D ) )
12:1,110:	|- (. A e. B ->. ( [. A / x ]. C C_ D <-> [. A / x ]. A. y ( y e. C -> y e. D ) ) ).
13:10,12:	|- (. A e. B ->. ( [. A / x ]. C C_ D <-> A. y ( y e. [_ A / x ]_ C -> y e. [_ A / x ]_ D ) ) ).
14::	|- ( [_ A / x ]_ C C_ [_ A / x ]_ D <-> A. y ( y e. [_ A / x ]_ C -> y e. [_ A / x ]_ D ) )
15:13,14:	|- (. A e. B ->. ( [. A / x ]. C C_ D <-> [_ A / x ]_ C C_ [_ A / x ]_ D ) ).
qed:15:	|- ( A e. B -> ( [. A / x ]. C C_ D <-> [_ A / x ]_ C C_ [_ A / x ]_ D ) )

(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( A e. B -> ( [. A / x ]. C C_ D <-> [_ A / x ]_ C C_ [_ A / x ]_ D ) )

Theorem

csbingVD 37278

Virtual deduction proof of csbingOLD 37212. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbingOLD 37212 is csbingVD 37278 without virtual deductions and was automatically derived from csbingVD 37278.

1::	|- (. A e. B ->. A e. B ).
2::	|- ( C i^i D ) = { y | ( y e. C /\ y e. D ) }
20:2:	|- A. x ( C i^i D ) = { y | ( y e. C /\ y e. D ) }
30:1,20:	|- (. A e. B ->. [. A / x ]. ( C i^i D ) = { y | ( y e. C /\ y e. D ) } ).
3:1,30:	|- (. A e. B ->. [_ A / x ]_ ( C i^i D ) = [_ A / x ]_ { y | ( y e. C /\ y e. D ) } ).
4:1:	|- (. A e. B ->. [_ A / x ]_ { y | ( y e. C /\ y e. D ) } = { y | [. A / x ]. ( y e. C /\ y e. D ) } ).
5:3,4:	|- (. A e. B ->. [_ A / x ]_ ( C i^i D ) = { y | [. A / x ]. ( y e. C /\ y e. D ) } ).
6:1:	|- (. A e. B ->. ( [. A / x ]. y e. C <-> y e. [_ A / x ]_ C ) ).
7:1:	|- (. A e. B ->. ( [. A / x ]. y e. D <-> y e. [_ A / x ]_ D ) ).
8:6,7:	|- (. A e. B ->. ( ( [. A / x ]. y e. C /\ [. A / x ]. y e. D ) <-> ( y e. [_ A / x ]_ C /\ y e. [_ A / x ]_ D ) ) ).
9:1:	|- (. A e. B ->. ( [. A / x ]. ( y e. C /\ y e. D ) <-> ( [. A / x ]. y e. C /\ [. A / x ]. y e. D ) ) ).
10:9,8:	|- (. A e. B ->. ( [. A / x ]. ( y e. C /\ y e. D ) <-> ( y e. [_ A / x ]_ C /\ y e. [_ A / x ]_ D ) ) ).
11:10:	|- (. A e. B ->. A. y ( [. A / x ]. ( y e. C /\ y e. D ) <-> ( y e. [_ A / x ]_ C /\ y e. [_ A / x ]_ D ) ) ).
12:11:	|- (. A e. B ->. { y | [. A / x ]. ( y e. C /\ y e. D ) } = { y | ( y e. [_ A / x ]_ C /\ y e. [_ A / x ]_ D ) } ).
13:5,12:	|- (. A e. B ->. [_ A / x ]_ ( C i^i D ) = { y | ( y e. [_ A / x ]_ C /\ y e. [_ A / x ]_ D ) } ).
14::	|- ( [_ A / x ]_ C i^i [_ A / x ]_ D ) = { y | ( y e. [_ A / x ]_ C /\ y e. [_ A / x ]_ D ) }
15:13,14:	|- (. A e. B ->. [_ A / x ]_ ( C i^i D ) = ( [_ A / x ]_ C i^i [_ A / x ]_ D ) ).
qed:15:	|- ( A e. B -> [_ A / x ]_ ( C i^i D ) = ( [_ A / x ]_ C i^i [_ A / x ]_ D ) )

(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( A e. B -> [_ A / x ]_ ( C i^i D ) = ( [_ A / x ]_ C i^i [_ A / x ]_ D ) )

Theorem

onfrALTlem5VD 37279*

Virtual deduction proof of onfrALTlem5 36909. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. onfrALTlem5 36909 is onfrALTlem5VD 37279 without virtual deductions and was automatically derived from onfrALTlem5VD 37279.

1::	|- a e. _V
2:1:	|- ( a i^i x ) e. _V
3:2:	|- ( [. ( a i^i x ) / b ]. b = (/) <-> ( a i^i x ) = (/) )
4:3:	|- ( -. [. ( a i^i x ) / b ]. b = (/) <-> -. ( a i^i x ) = (/) )
5::	|- ( ( a i^i x ) =/= (/) <-> -. ( a i^i x ) = (/) )
6:4,5:	|- ( -. [. ( a i^i x ) / b ]. b = (/) <-> ( a i^i x ) =/= (/) )
7:2:	|- ( -. [. ( a i^i x ) / b ]. b = (/) <-> [. ( a i^i x ) / b ]. -. b = (/) )
8::	|- ( b =/= (/) <-> -. b = (/) )
9:8:	|- A. b ( b =/= (/) <-> -. b = (/) )
10:2,9:	|- ( [. ( a i^i x ) / b ]. b =/= (/) <-> [. ( a i^i x ) / b ]. -. b = (/) )
11:7,10:	|- ( -. [. ( a i^i x ) / b ]. b = (/) <-> [. ( a i^i x ) / b ]. b =/= (/) )
12:6,11:	|- ( [. ( a i^i x ) / b ]. b =/= (/) <-> ( a i^i x ) =/= (/) )
13:2:	|- ( [. ( a i^i x ) / b ]. b C_ ( a i^i x ) <-> ( a i^i x ) C_ ( a i^i x ) )
14:12,13:	|- ( ( [. ( a i^i x ) / b ]. b C_ ( a i^i x ) /\ [. ( a i^i x ) / b ]. b =/= (/) ) <-> ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) )
15:2:	|- ( [. ( a i^i x ) / b ]. ( b C_ ( a i^i x ) /\ b =/= (/) ) <-> ( [. ( a i^i x ) / b ]. b C_ ( a i^i x ) /\ [. ( a i^i x ) / b ]. b =/= (/) ) )
16:15,14:	|- ( [. ( a i^i x ) / b ]. ( b C_ ( a i^i x ) /\ b =/= (/) ) <-> ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) )
17:2:	|- [_ ( a i^i x ) / b ]_ ( b i^i y ) = ( [_ ( a i^i x ) / b ]_ b i^i [_ ( a i^i x ) / b ]_ y )
18:2:	|- [_ ( a i^i x ) / b ]_ b = ( a i^i x )
19:2:	|- [_ ( a i^i x ) / b ]_ y = y
20:18,19:	|- ( [_ ( a i^i x ) / b ]_ b i^i [_ ( a i^i x ) / b ]_ y ) = ( ( a i^i x ) i^i y )
21:17,20:	|- [_ ( a i^i x ) / b ]_ ( b i^i y ) = ( ( a i^i x ) i^i y )
22:2:	|- ( [. ( a i^i x ) / b ]. ( b i^i y ) = (/) <-> [_ ( a i^i x ) / b ]_ ( b i^i y ) = [_ ( a i^i x ) / b ]_ (/) )
23:2:	|- [_ ( a i^i x ) / b ]_ (/) = (/)
24:21,23:	|- ( [_ ( a i^i x ) / b ]_ ( b i^i y ) = [_ ( a i^i x ) / b ]_ (/) <-> ( ( a i^i x ) i^i y ) = (/) )
25:22,24:	|- ( [. ( a i^i x ) / b ]. ( b i^i y ) = (/) <-> ( ( a i^i x ) i^i y ) = (/) )
26:2:	|- ( [. ( a i^i x ) / b ]. y e. b <-> y e. ( a i^i x ) )
27:25,26:	|- ( ( [. ( a i^i x ) / b ]. y e. b /\ [. ( a i^i x ) / b ]. ( b i^i y ) = (/) ) <-> ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) )
28:2:	|- ( [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> ( [. ( a i^i x ) / b ]. y e. b /\ [. ( a i^i x ) / b ]. ( b i^i y ) = (/) ) )
29:27,28:	|- ( [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) )