P. 336 of Theorem List - Metamath Proof Explorer

Type

Label

Description

Statement

Theorem

unipwr 33501

A class is a subclass of the union of its power class. This theorem is the right-to-left subclass lemma of unipw 4687. The proof of this theorem was automatically generated from unipwrVD 33500 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 33502

Virtual deduction proof of sstrALT2 33503. (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 33503

Virtual deduction proof of sstr 3497, transitivity of subclasses, Theorem 6 of [Suppes] p. 23. This theorem was automatically generated from sstrALT2VD 33502 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 33504

Virtual deduction proof of suctrALT2 33505. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( Tr A -> Tr suc A )

Theorem

suctrALT2 33505

Virtual deduction proof of suctr 4951. The sucessor of a transitive class is transitive. This proof was generated automatically from the virtual deduction proof suctrALT2VD 33504 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 33506*

Virtual deduction proof of elex2 3107. (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 33507*

Virtual deduction proof of elex22 3108. (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 33508*

Virtual deduction proof of eqsbc3r 3375. (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 33509*

Virtual deduction proof of zfregs2 8167. (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 33510

Virtual deduction proof of tpid3g 4130. (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 33511*

Virtual deduction proof of en3lplem1 8034. (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 33512*

Virtual deduction proof of en3lplem2 8035. (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 33513

Virtual deduction proof of en3lp 8036. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

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

21.27.8  Theorems proved using Virtual Deduction with mmj2 assistance

Theorem

simplbi2VD 33514

Virtual deduction proof of simplbi2 625. 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 33437	|- ( ( ps /\ ch ) -> ph )
qed:3,?: e0a 33437	|- ( ps -> ( ch -> ph ) )

The proof of simplbi2 625 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 33515

Virtual deduction proof of 3ornot23 33146. 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 33281	|- (. ( -. ph /\ -. ps ) ->. -. ph ).
4:1,?: e1a 33281	|- (. ( -. ph /\ -. ps ) ->. -. ps ).
5:3,4,?: e11 33342	|- (. ( -. ph /\ -. ps ) ->. -. ( ph \/ ps ) ).
6:2,?: e2 33285	|- (. ( -. ph /\ -. ps ) ,. ( ch \/ ph \/ ps ) ->. ( ch \/ ( ph \/ ps ) ) ).
7:5,6,?: e12 33389	|- (. ( -. 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 33516

Virtual deduction proof of orbi1r 33147. 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 33285	|- (. ( ph <-> ps ) ,. ( ch \/ ph ) ->. ( ph \/ ch ) ).
4:1,3,?: e12 33389	|- (. ( ph <-> ps ) ,. ( ch \/ ph ) ->. ( ps \/ ch ) ).
5:4,?: e2 33285	|- (. ( ph <-> ps ) ,. ( ch \/ ph ) ->. ( ch \/ ps ) ).
6:5:	|- (. ( ph <-> ps ) ->. ( ( ch \/ ph ) -> ( ch \/ ps ) ) ).
7::	|- (. ( ph <-> ps ) ,. ( ch \/ ps ) ->. ( ch \/ ps ) ).
8:7,?: e2 33285	|- (. ( ph <-> ps ) ,. ( ch \/ ps ) ->. ( ps \/ ch ) ).
9:1,8,?: e12 33389	|- (. ( ph <-> ps ) ,. ( ch \/ ps ) ->. ( ph \/ ch ) ).
10:9,?: e2 33285	|- (. ( ph <-> ps ) ,. ( ch \/ ps ) ->. ( ch \/ ph ) ).
11:10:	|- (. ( ph <-> ps ) ->. ( ( ch \/ ps ) -> ( ch \/ ph ) ) ).
12:6,11,?: e11 33342	|- (. ( 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 33517

Virtual deduction proof of bitr3 33148. 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 33281	|- (. ( ph <-> ps ) ->. ( ps <-> ph ) ).
3::	|- (. ( ph <-> ps ) ,. ( ph <-> ch ) ->. ( ph <-> ch ) ).
4:3,?: e2 33285	|- (. ( ph <-> ps ) ,. ( ph <-> ch ) ->. ( ch <-> ph ) ).
5:2,4,?: e12 33389	|- (. ( 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 33518

Virtual deduction proof of 3orbi123 33149. 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 33281	|- (. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ->. ( ph <-> ps ) ).
3:1,?: e1a 33281	|- (. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ->. ( ch <-> th ) ).
4:1,?: e1a 33281	|- (. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ->. ( ta <-> et ) ).
5:2,3,?: e11 33342	|- (. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ->. ( ( ph \/ ch ) <-> ( ps \/ th ) ) ).
6:5,4,?: e11 33342	|- (. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ->. ( ( ( ph \/ ch ) \/ ta ) <-> ( ( ps \/ th ) \/ et ) ) ).
7:?:	|- ( ( ( ph \/ ch ) \/ ta ) <-> ( ph \/ ch \/ ta ) )
8:6,7,?: e10 33348	|- (. ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) ->. ( ( ph \/ ch \/ ta ) <-> ( ( ps \/ th ) \/ et ) ) ).
9:?:	|- ( ( ( ps \/ th ) \/ et ) <-> ( ps \/ th \/ et ) )
10:8,9,?: e10 33348	|- (. ( ( 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 33519

Virtual deduction proof of sbc3orgOLD 33171. 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 33281	|- (. 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 33348	|- (. A e. B ->. [. A / x ]. ( ( ( ph \/ ps ) \/ ch ) <-> ( ph \/ ps \/ ch ) ) ).
4:1,33,?: e11 33342	|- (. A e. B ->. ( [. A / x ]. ( ( ph \/ ps ) \/ ch ) <-> [. A / x ]. ( ph \/ ps \/ ch ) ) ).
5:2,4,?: e11 33342	|- (. A e. B ->. ( [. A / x ]. ( ph \/ ps \/ ch ) <-> ( [. A / x ]. ( ph \/ ps ) \/ [. A / x ]. ch ) ) ).
6:1,?: e1a 33281	|- (. A e. B ->. ( [. A / x ]. ( ph \/ ps ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps ) ) ).
7:6,?: e1a 33281	|- (. A e. B ->. ( ( [. A / x ]. ( ph \/ ps ) \/ [. A / x ]. ch ) <-> ( ( [. A / x ]. ph \/ [. A / x ]. ps ) \/ [. A / x ]. ch ) ) ).
8:5,7,?: e11 33342	|- (. 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 33348	|- (. 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 33520*

Virtual deduction proof of alrim3con13v 33172. 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 33285	|- (. ( ph -> A. x ph ) ,. ( ps /\ ph /\ ch ) ->. ps ).
4:2,?: e2 33285	|- (. ( ph -> A. x ph ) ,. ( ps /\ ph /\ ch ) ->. ph ).
5:2,?: e2 33285	|- (. ( ph -> A. x ph ) ,. ( ps /\ ph /\ ch ) ->. ch ).
6:1,4,?: e12 33389	|- (. ( ph -> A. x ph ) ,. ( ps /\ ph /\ ch ) ->. A. x ph ).
7:3,?: e2 33285	|- (. ( ph -> A. x ph ) ,. ( ps /\ ph /\ ch ) ->. A. x ps ).
8:5,?: e2 33285	|- (. ( ph -> A. x ph ) ,. ( ps /\ ph /\ ch ) ->. A. x ch ).
9:7,6,8,?: e222 33290	|- (. ( ph -> A. x ph ) ,. ( ps /\ ph /\ ch ) ->. ( A. x ps /\ A. x ph /\ A. x ch ) ).
10:9,?: e2 33285	|- (. ( 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 33521

Virtual deduction proof of exbir 33087. 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 33389	|- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) , ( ph /\ ps ) ->. ( ch <-> th ) ).
6:3,5,?: e32 33423	|- (. ( ( 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 33281	|- (. ( ( 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 33522

Virtual deduction proof of exbiri 622. 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 33348	|- (. ph ->. ( ps -> ( ch <-> th ) ) ).
6:3,5,?: e21 33395	|- (. ph ,. ps ->. ( ch <-> th ) ).
7:4,6,?: e32 33423	|- (. 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 33523*

Virtual deduction proof of rspsbc2 33173. 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 33413	|- (. 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 33413	|- (. 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 33420	|- (. 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 33524

Virtual deduction proof of 3impexp 33088. 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 33348	|- (. ( ( ph /\ ps /\ ch ) -> th ) ->. ( ( ( ph /\ ps ) /\ ch ) -> th ) ).
4:3,?: e1a 33281	|- (. ( ( ph /\ ps /\ ch ) -> th ) ->. ( ( ph /\ ps ) -> ( ch -> th ) ) ).
5:4,?: e1a 33281	|- (. ( ( 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 33281	|- (. ( ph -> ( ps -> ( ch -> th ) ) ) ->. ( ( ph /\ ps ) -> ( ch -> th ) ) ).
9:8,?: e1a 33281	|- (. ( ph -> ( ps -> ( ch -> th ) ) ) ->. ( ( ( ph /\ ps ) /\ ch ) -> th ) ).
10:2,9,?: e01 33345	|- (. ( ph -> ( ps -> ( ch -> th ) ) ) ->. ( ( ph /\ ps /\ ch ) -> th ) ).
11:10:	|- ( ( ph -> ( ps -> ( ch -> th ) ) ) -> ( ( ph /\ ps /\ ch ) -> th ) )
qed:6,11,?: e00 33433	|- ( ( ( 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 33525

Virtual deduction proof of 3impexpbicom 33089. 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 33348	|- (. ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) ->. ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) ).
4:3,?: e1a 33281	|- (. ( ( 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 33281	|- (. ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ->. ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) ).
8:7,2,?: e10 33348	|- (. ( 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 33433	|- ( ( ( 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 33526

Virtual deduction proof of 3impexpbicomi 33090. 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 33437	|- ( 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

sbcoreleleqVD 33527*

Virtual deduction proof of sbcoreleleq 33174. 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 33281	|- (. A e. B ->. ( [. A / y ]. x e. y <-> x e. A ) ).
3:1,?: e1a 33281	|- (. A e. B ->. ( [. A / y ]. y e. x <-> A e. x ) ).
4:1,?: e1a 33281	|- (. A e. B ->. ( [. A / y ]. x = y <-> x = A ) ).
5:2,3,4,?: e111 33328	|- (. 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 33281	|- (. 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 33342	|- (. 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 33528*

Virtual deduction proof of nfra2 2830. 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 33433	|- ( 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 33437	|- ( 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 33433	|- ( 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 33529*

Virtual deduction proof of tratrb 33175. 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 33281	|- (. ( 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 33281	|- (. ( 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 33281	|- (. ( 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 33285	|- (. ( 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 33285	|- (. ( 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 33310	|- (. ( 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 33307	|- (. ( 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 33289	|- (. ( 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 33392	|- (. ( 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 33420	|- (. ( 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 33420	|- (. ( 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 33392	|- (. ( 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 33281	|- (. ( 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 33310	|- (. ( 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 33285	|- (. ( 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 33389	|- (. ( 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 33290	|- (. ( 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 33437	|- ( ( 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 33437	|- ( ( 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 33281	|- (. ( 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 33530

Virtual deduction proof of al2im 1622. 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 33281	|- (. 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 33348	|- (. 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 33531

Virtual deduction proof of syl5imp 33150. 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 33281	|- (. ( ph -> ( ps -> ch ) ) ->. ( ps -> ( ph -> ch ) ) ).
3::	|- (. ( ph -> ( ps -> ch ) ) ,. ( th -> ps ) ->. ( th -> ps ) ).
4:3,2,?: e21 33395	|- (. ( ph -> ( ps -> ch ) ) ,. ( th -> ps ) ->. ( th -> ( ph -> ch ) ) ).
5:4,?: e2 33285	|- (. ( 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 33532

Virtual deduction proof of idiALT 33086. 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 33437	|- ph

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

|- ph   =>    |- ph

Theorem

ancomstVD 33533

Closed form of ancoms 453. 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 33437	|- ( ( ( ph /\ ps ) -> ch ) <-> ( ( ps /\ ph ) -> ch ) )

The proof of ancomst 452 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 33534*

Quantification restricted to a subclass for two quantifiers. ssralv 3549 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 33169 is ssralv2VD 33534 without virtual deductions and was automatically derived from ssralv2VD 33534.

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 33535

An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 4890 using the Axiom of Regularity indirectly through dford2 8040. 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 33176 is ordelordALTVD 33535 without virtual deductions and was automatically derived from ordelordALTVD 33535 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 33536

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 3634 is equncomVD 33536 without virtual deductions and was automatically derived from equncomVD 33536.

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 33537

Inference form of equncom 3634. 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 3635 is equncomiVD 33537 without virtual deductions and was automatically derived from equncomiVD 33537.

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 33538

A set belongs to its successor. Alternate proof of sucid 4947. 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 33539 is sucidALTVD 33538 without virtual deductions and was automatically derived from sucidALTVD 33538. 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 4874, 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 8040.

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 33539

A set belongs to its successor. This proof was automatically derived from sucidALTVD 33538 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 33540

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 4947 is sucidVD 33540 without virtual deductions and was automatically derived from sucidVD 33540.

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 33541

Implication form of imbi12i 326. 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 322 is imbi12VD 33541 without virtual deductions and was automatically derived from imbi12VD 33541.

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 33542

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 33158 is imbi13VD 33542 without virtual deductions and was automatically derived from imbi13VD 33542.

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 33543

Distribution of class substitution over a left-nested implication. Similar to sbcimg 3355. 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 33177 is sbcim2gVD 33543 without virtual deductions and was automatically derived from sbcim2gVD 33543.

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 33544

Implication form of sbcbiiOLD 3374. 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 33178 is sbcbiVD 33544 without virtual deductions and was automatically derived from sbcbiVD 33544.

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 33545*

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 33179 is trsbcVD 33545 without virtual deductions and was automatically derived from trsbcVD 33545.

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 33546*

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 33180 is truniALTVD 33546 without virtual deductions and was automatically derived from truniALTVD 33546.

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 33547

Non-virtual deduction form of e33 33399. 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 33159 is ee33VD 33547 without virtual deductions and was automatically derived from ee33VD 33547.

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 33548*

The intersection of a class of transitive sets is transitive. Virtual deduction proof of trintALT 33549. 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 33549 is trintALTVD 33548 without virtual deductions and was automatically derived from trintALTVD 33548.

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 33549*

The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. trintALT 33549 is an alternative proof of trint 4545. trintALT 33549 is trintALTVD 33548 without virtual deductions and was automatically derived from trintALTVD 33548 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 33550

The first equality of Exercise 13 of [TakeutiZaring] p. 22. Virtual deduction proof of undif3 3744. 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 3744 is undif3VD 33550 without virtual deductions and was automatically derived from undif3VD 33550.

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 33551

Virtual deduction proof of sbcssg 3925. 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 3925 is sbcssgVD 33551 without virtual deductions and was automatically derived from sbcssgVD 33551.

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 33552

Virtual deduction proof of csbingOLD 3847. 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 3847 is csbingVD 33552 without virtual deductions and was automatically derived from csbingVD 33552.

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 33553*

Virtual deduction proof of onfrALTlem5 33182. 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 33182 is onfrALTlem5VD 33553 without virtual deductions and was automatically derived from onfrALTlem5VD 33553.

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 ) = (/) ) )
30:29:	|- A. y ( [. ( 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 ) = (/) ) )
31:30:	|- ( E. y [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> E. y ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) )
32::	|- ( E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) <-> E. y ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) )
33:31,32:	|- ( E. y [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) )
34:2:	|- ( E. y [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> [. ( a i^i x ) / b ]. E. y ( y e. b /\ ( b i^i y ) = (/) ) )
35:33,34:	|- ( [. ( a i^i x ) / b ]. E. y ( y e. b /\ ( b i^i y ) = (/) ) <-> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) )
36::	|- ( E. y e. b ( b i^i y ) = (/) <-> E. y ( y e. b /\ ( b i^i y ) = (/) ) )
37:36:	|- A. b ( E. y e. b ( b i^i y ) = (/) <-> E. y ( y e. b /\ ( b i^i y ) = (/) ) )
38:2,37:	|- ( [. ( a i^i x ) / b ]. E. y e. b ( b i^i y ) = (/) <-> [. ( a i^i x ) / b ]. E. y ( y e. b /\ ( b i^i y ) = (/) ) )
39:35,38:	|- ( [. ( a i^i x ) / b ]. E. y e. b ( b i^i y ) = (/) <-> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) )
40:16,39:	|- ( ( [. ( a i^i x ) / b ]. ( b C_ ( a i^i x ) /\ b =/= (/) ) -> [. ( a i^i x ) / b ]. E. y e. b ( b i^i y ) = (/) ) <-> ( ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) -> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) ) )
41:2:	|- ( [. ( a i^i x ) / b ]. ( ( b C_ ( a i^i x ) /\ b =/= (/) ) -> E. y e. b ( b i^i y ) = (/) ) <-> ( [. ( a i^i x ) / b ]. ( b C_ ( a i^i x ) /\ b =/= (/) ) -> [. ( a i^i x ) / b ]. E. y e. b ( b i^i y ) = (/) ) )
qed:40,41:	|- ( [. ( a i^i x ) / b ]. ( ( b C_ ( a i^i x ) /\ b =/= (/) ) -> E. y e. b ( b i^i y ) = (/) ) <-> ( ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) -> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) ) )

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

|- ( [. ( a i^i x ) / b ]. ( ( b C_ ( a i^i x ) /\ b =/= (/) ) -> E. y e. b ( b i^i y ) = (/) ) <-> ( ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) -> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) ) )

Theorem

onfrALTlem4VD 33554*

Virtual deduction proof of onfrALTlem4 33183. 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. onfrALTlem4 33183 is onfrALTlem4VD 33554 without virtual deductions and was automatically derived from onfrALTlem4VD 33554.

1::	|- y e. _V
2:1:	|- ( [. y / x ]. ( a i^i x ) = (/) <-> [_ y