HomeHome Metamath Proof Explorer
Theorem List (p. 346 of 382)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-25961)
  Hilbert Space Explorer  Hilbert Space Explorer
(25962-27486)
  Users' Mathboxes  Users' Mathboxes
(27487-38122)
 

Theorem List for Metamath Proof Explorer - 34501-34600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
21.29.4.15  Some Principia Mathematica proofs

References are made to the second edition (1927, reprinted 1963) of Principia Mathematica, Vol. 1. Theorems are referred to in the form "PM*xx.xx".

 
Theoremstdpc5t 34501 Closed form of stdpc5 1909. (Possible to place it before 19.21t 1905 and use it to prove 19.21t 1905). (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
 |-  ( F/ x ph  ->  ( A. x ( ph  ->  ps )  ->  ( ph  ->  A. x ps )
 ) )
 
Theorembj-stdpc5 34502 More direct proof of stdpc5 1909. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
 |-  F/ x ph   =>    |-  ( A. x (
 ph  ->  ps )  ->  ( ph  ->  A. x ps )
 )
 
Theorem2stdpc5 34503 A double stdpc5 1909 (one direction of PM*11.3). See also 2stdpc4 2096 and 19.21vv 31443. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
 |-  F/ x ph   &    |-  F/ y ph   =>    |-  ( A. x A. y (
 ph  ->  ps )  ->  ( ph  ->  A. x A. y ps ) )
 
Theorembj-19.21t 34504 Proof of 19.21t 1905 from stdpc5t 34501. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
 |-  ( F/ x ph  ->  ( A. x ( ph  ->  ps )  <->  ( ph  ->  A. x ps ) ) )
 
Theoremexlimii 34505 Inference associated with exlimi 1913. Inferring a theorem when it is implied by an antecedent which may be true. (Contributed by BJ, 15-Sep-2018.)
 |-  F/ x ps   &    |-  ( ph  ->  ps )   &    |-  E. x ph   =>    |-  ps
 
Theoremax11-pm 34506 Proof of ax-11 1843 similar to PM's proof of alcom 1846 (PM*11.2). For a proof closer to PM's proof, see ax11-pm2 34510. Axiom ax-11 1843 is used in the proof only through nfa2 1954. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
 |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Theoremax6er 34507 Another form of ax6e 2003. ( Could be placed right after ax6e 2003). (Contributed by BJ, 15-Sep-2018.)
 |-  E. x  y  =  x
 
Theoremexlimiieq1 34508 Inferring a theorem when it is implied by an equality which may be true. (Contributed by BJ, 30-Sep-2018.)
 |-  F/ x ph   &    |-  ( x  =  y  ->  ph )   =>    |-  ph
 
Theoremexlimiieq2 34509 Inferring a theorem when it is implied by an equality which may be true. (Contributed by BJ, 15-Sep-2018.) (Revised by BJ, 30-Sep-2018.)
 |-  F/ y ph   &    |-  ( x  =  y  ->  ph )   =>    |-  ph
 
Theoremax11-pm2 34510* Proof of ax-11 1843 from the standard axioms of predicate calculus, similar to PM's proof of alcom 1846 (PM*11.2). This proof requires that  x and  y be distinct. Axiom ax-11 1843 is used in the proof only through nfal 1948, nfsb 2185, sbal 2207, sb8 2168. See also ax11-pm 34506. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
 |-  ( A. x A. y ph  ->  A. y A. x ph )
 
21.29.4.16  Lemmas for substitution
 
Theorembj-sbf3 34511 Substitution has no effect on a bound variabe (existential quantifier case); see sbf2 2124. (Contributed by BJ, 2-May-2019.)
 |-  ( [ y  /  x ] E. x ph  <->  E. x ph )
 
Theorembj-sbf4 34512 Substitution has no effect on a bound variabe (non-freeness case); see sbf2 2124. (Contributed by BJ, 2-May-2019.)
 |-  ( [ y  /  x ] F/ x ph  <->  F/ x ph )
 
Theorembj-sbnf 34513* Move non-free predicate in and out of substitution; see sbal 2207 and sbex 2208. (Contributed by BJ, 2-May-2019.)
 |-  ( [ z  /  y ] F/ x ph  <->  F/ x [ z  /  y ] ph )
 
21.29.4.17  Existential uniqueness
 
Theorembj-eu3f 34514* Version of eu3v 2313 where the dv condition is replaced with a non-freeness hypothesis. This is a "backup" of a theorem that used to be in the main part with label "eu3" and was deprecated in favor of eu3v 2313. (Contributed by NM, 8-Jul-1994.) (Proof shortened by BJ, 31-May-2019.)
 |-  F/ y ph   =>    |-  ( E! x ph  <->  ( E. x ph  /\  E. y A. x ( ph  ->  x  =  y ) ) )
 
Theorembj-eumo0 34515* Existential uniqueness implies "at most one." Used to be in the main part and deprecated in favor of eumo 2314 and mo2 2294. (Contributed by NM, 8-Jul-1994.) (Revised by BJ, 8-Jun-2019.)
 |-  F/ y ph   =>    |-  ( E! x ph  ->  E. y A. x ( ph  ->  x  =  y ) )
 
21.29.5  Set theory
 
21.29.5.1  Eliminability of class terms

In this section, we give a sketch of the proof of the Eliminability Theorem for class terms in an extensional set theory where quantification occurs only over set variables.

Eliminability of class variables using the $a-statements ax-ext 2435, df-clab 2443, df-cleq 2449, df-clel 2452 is an easy result, proved for instance in Appendix X of Azriel Levy, Basic Set Theory, Dover Publications, 2002. Note that viewed from the set.mm axiomatization, it is a metatheorem not formalizable is set.mm. It states: every formula in the language of FOL +  e. + class terms, but without class variables is provably equivalent (over {FOL, ax-ext 2435, df-clab 2443, df-cleq 2449, df-clel 2452 }) to a formula in the language of FOL + 
e. (that is, without class terms).

The proof goes by induction on the complexity of the formula (see op. cit. for details). The base case is that of atomic formulas. The atomic formulas containing class terms are of one of the following forms: for equality,  x  =  { y  |  ph },  { x  |  ph }  =  y,  {
x  |  ph }  =  { y  |  ps }, and for membership,  y  e.  { x  |  ph },  { x  |  ph }  e.  y,  { x  |  ph }  e.  {
y  |  ps }. These cases are dealt with by eliminable1 34516 and the following theorems of this section, which are special instances of df-clab 2443, dfcleq 2450 (proved from {FOL, ax-ext 2435, df-cleq 2449 }), and df-clel 2452. Indeed, denote by (i) the formula proved by "eliminablei". One sees that the RHS of (1) has no class terms, the RHS's of (2x) have only class terms of the form dealt with by (1), and the RHS's of (3x) have only class terms of the forms dealt with by (1) and (2a). Note that in order to prove eliminable2a 34517, eliminable2b 34518 and eliminable3a 34520, we need to substitute a class variable for a setvar variable. This is possible because setvars are class terms: this is the content of the syntactic theorem cv 1394, which is used in these proofs (this does not appear in the html pages but it is in the set.mm file and you can check it using the Metamath program).

The induction step relies on the fact that any formula is a FOL-combination of atomic formulas, so if one found equivalents for all atomic formulas constituting the formula, then the same FOL-combination of these equivalents will be equivalent to the original formula.

Note that one has a slightly more precise result: if the original formula has only class terms appearing in atomic formulas of the form  y  e.  { x  |  ph }, then df-clab 2443 is sufficient (over FOL) to eliminate class terms, and if the original formula has only class terms appearing in atomic formulas of the form  y  e.  { x  |  ph } and equalities, then df-clab 2443, ax-ext 2435 and df-cleq 2449 are sufficient (over FOL) to eliminate class terms.

To prove that { df-clab 2443, df-cleq 2449, df-clel 2452 } provides a definitional extension of {FOL, ax-ext 2435 }, one needs to prove the above Eliminability Theorem, which compares the expressive powers of the languages with and without class terms, and the Conservativity Theorem, which compares the deductive powers when one adds { df-clab 2443, df-cleq 2449, df-clel 2452 }. It states that a formula without class terms is provable in one axiom system if and only if it is provable in the other, and that this remains true when one adds further definitions to {FOL, ax-ext 2435 } . It is also proved in op. cit. The proof is more difficult, since one has to construct for each proof of a statement without class terms, an associated proof not using { df-clab 2443, df-cleq 2449, df-clel 2452 }. It involves a careful case study on the structure of the proof tree.

 
Theoremeliminable1 34516 A theorem used to prove the base case of the Eliminability Theorem (see section comment). (Contributed by BJ, 19-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 y  e.  { x  |  ph }  <->  [ y  /  x ] ph )
 
Theoremeliminable2a 34517* A theorem used to prove the base case of the Eliminability Theorem (see section comment). (Contributed by BJ, 19-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( x  =  { y  |  ph }  <->  A. z ( z  e.  x  <->  z  e.  { y  |  ph } ) )
 
Theoremeliminable2b 34518* A theorem used to prove the base case of the Eliminability Theorem (see section comment). (Contributed by BJ, 19-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( { x  |  ph }  =  y 
 <-> 
 A. z ( z  e.  { x  |  ph
 } 
 <->  z  e.  y ) )
 
Theoremeliminable2c 34519* A theorem used to prove the base case of the Eliminability Theorem (see section comment). (Contributed by BJ, 19-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( { x  |  ph }  =  { y  |  ps }  <->  A. z ( z  e. 
 { x  |  ph }  <-> 
 z  e.  { y  |  ps } ) )
 
Theoremeliminable3a 34520* A theorem used to prove the base case of the Eliminability Theorem (see section comment). (Contributed by BJ, 19-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( { x  |  ph }  e.  y 
 <-> 
 E. z ( z  =  { x  |  ph
 }  /\  z  e.  y ) )
 
Theoremeliminable3b 34521* A theorem used to prove the base case of the Eliminability Theorem (see section comment). (Contributed by BJ, 19-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( { x  |  ph }  e.  { y  |  ps }  <->  E. z ( z  =  { x  |  ph } 
 /\  z  e.  {
 y  |  ps }
 ) )
 
Theorembj-termab 34522* Every class can be written as (is equal to) a class abstraction. cvjust 2451 is a special instance of it, but the present proof does not require ax-13 2000, contrary to cvjust 2451. This theorem requires ax-ext 2435, df-clab 2443, df-cleq 2449, df-clel 2452, but to prove that any specific class term not containing class variables is a setvar or can be written as (is equal to) a class abstraction does not require these $a-statements. This last fact is a metatheorem, consequence of the fact that the only $a-statements with typecode  class are cv 1394, cab 2442 and statements corresponding to defined class constructors. UPDATE: it is (almost) abid2 2597 and bj-abid2 34469, though the present proof is shorter than a proof from bj-abid2 34469 and eqcomi 2470 (and is shorter than the proof of either) ; plus, it is of the same form as cvjust 2451 and such a basic statement deserves to be present in both forms. Note that bj-termab 34522 shortens the proof of abid2 2597, and shortens five proofs by a total of 72 bytes. Move it to main as "abid1"? (Contributed by BJ, 21-Oct-2019.) (Proof modification is discouraged.)
 |-  A  =  { x  |  x  e.  A }
 
21.29.5.2  Classes without extensionality

A few results about classes can be proved without using ax-ext 2435. One could move all theorems from cab 2442 to df-clel 2452 (except for dfcleq 2450 and cvjust 2451) in a subsection "Classes" before the subsection on the axiom of extensionality, together with the theorems below. In that subsection, the last statement should be df-cleq 2449.

Note that without ax-ext 2435, the $a-statements df-clab 2443, df-cleq 2449, and df-clel 2452 are no longer eliminable (see previous section) (but PROBABLY are still conservative). This is not a reason not to study what is provable with them but without ax-ext 2435, in order to gauge their strengths more precisely.

Before that subsection, a subsection "The membership predicate" could group the statements with  e. that are currently in the FOL part (including wcel 1819, wel 1820, ax-8 1821, ax-9 1823).

 
Theorembj-eleq1w 34523 Weaker version of eleq1 2529 (but more general than elequ1 1822) not depending on ax-ext 2435 (nor ax-12 1855 nor df-cleq 2449). Remark: this can also be done with eleq1i 2534, eqeltri 2541, eqeltrri 2542, eleq1a 2540, eleq1d 2526, eqeltrd 2545, eqeltrrd 2546, eqneltrd 2566, eqneltrrd 2567, nelneq 2574. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
 |-  ( x  =  y  ->  ( x  e.  A  <->  y  e.  A ) )
 
Theorembj-eleq2w 34524 Weaker version of eleq2 2530 (but more general than elequ2 1824) not depending on ax-ext 2435 (nor ax-12 1855 nor df-cleq 2449). (Contributed by BJ, 29-Sep-2019.) (Proof modification is discouraged.)
 |-  ( x  =  y  ->  ( A  e.  x  <->  A  e.  y
 ) )
 
Theorembj-clelsb3 34525* Remove dependency on ax-ext 2435 (and df-cleq 2449) from clelsb3 2578. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
 |-  ( [ x  /  y ] y  e.  A  <->  x  e.  A )
 
Theorembj-hblem 34526* Remove dependency on ax-ext 2435 (and df-cleq 2449) from hblem 2580. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
 |-  (
 y  e.  A  ->  A. x  y  e.  A )   =>    |-  ( z  e.  A  ->  A. x  z  e.  A )
 
Theorembj-nfcjust 34527* Remove dependency on ax-ext 2435 (and df-cleq 2449 and ax-13 2000) from nfcjust 2606. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
 |-  ( A. y F/ x  y  e.  A  <->  A. z F/ x  z  e.  A )
 
Theorembj-nfcrii 34528* Remove dependency on ax-ext 2435 (and df-cleq 2449) from nfcrii 2611. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
 |-  F/_ x A   =>    |-  ( y  e.  A  ->  A. x  y  e.  A )
 
Theorembj-nfcri 34529* Remove dependency on ax-ext 2435 (and df-cleq 2449) from nfcri 2612. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
 |-  F/_ x A   =>    |- 
 F/ x  y  e.  A
 
Theorembj-nfnfc 34530 Remove dependency on ax-ext 2435 (and df-cleq 2449) from nfnfc 2628. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
 |-  F/_ x A   =>    |- 
 F/ x F/_ y A
 
Theorembj-vexwt 34531 Closed form of bj-vexw 34532. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.) Use bj-vexwvt 34533 instead when sufficient. (New usage is discouraged.)
 |-  ( A. x ph  ->  y  e.  { x  |  ph } )
 
Theorembj-vexw 34532 If  ph is a theorem, then any set belongs to the class  { x  | 
ph }. Therefore,  { x  | 
ph } is "a" universal class.

This is the closest one can get to defining a universal class, or proving vex 3112, without using ax-ext 2435. Note that this theorem has no dv condition and does not use df-clel 2452 nor df-cleq 2449 either: only first-order logic and df-clab 2443.

Without ax-ext 2435, one cannot define "the" universal class, since one could not prove for instance the justification theorem  { x  | T.  }  =  {
y  | T.  }. Indeed, in order to prove any equality of classes, one needs df-cleq 2449, which has ax-ext 2435 as a hypothesis. Therefore, the classes  { x  | T.  },  { y  |  (
ph  ->  ph ) },  { z  |  ( A. t t  =  t  ->  A. t
t  =  t ) } and countless others are all universal classes whose equality one cannot prove without ax-ext 2435. See also bj-issetw 34537.

A version with a dv condition between  x and  y and not requiring ax-13 2000 is proved as bj-vexwv 34534, while the degenerate instance is a simple consequence of abid 2444. (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.) Use bj-vexwv 34534 instead when sufficient. (New usage is discouraged.)

 |-  ph   =>    |-  y  e.  { x  |  ph }
 
Theorembj-vexwvt 34533* Closed form of bj-vexwv 34534 and version of bj-vexwt 34531 with a dv condition, which does not require ax-13 2000. (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.)
 |-  ( A. x ph  ->  y  e.  { x  |  ph } )
 
Theorembj-vexwv 34534* Version of bj-vexw 34532 with a dv condition, which does not require ax-13 2000. The degenerate instance of bj-vexw 34532 is a simple consequence of abid 2444. (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.)
 |-  ph   =>    |-  y  e.  { x  |  ph }
 
Theorembj-denotes 34535* This would be the justification for the definition of the unary predicate "E!" by  |-  ( E!  A  <->  E. x x  =  A ) which could be interpreted as " A exists" or " A denotes". It is interesting that this justification can be proved without ax-ext 2435 nor df-cleq 2449 (but of course using df-clab 2443 and df-clel 2452). Once extensionality is postulated, then isset 3113 will prove that "existing" (as a set) is equivalent to being a member of a class.

Note that there is no dv condition on  x ,  y but the theorem does not depend on ax-13 2000. Actually, the proof depends only on ax-1--7 and sp 1860.

The symbol "E!" was chosen to be reminiscent of the analogous predicate in (inclusive or non-inclusive) free logic, which deals with the possibility of non-existent objects. This analogy should not be taken too far, since here there are no equality axioms for classes: they are derived from ax-ext 2435 (e.g. eqid 2457). In particular, one cannot even prove  |-  E. x x  =  A => 
|-  A  =  A.

With ax-ext 2435, the present theorem is obvious from cbvexv 2025 and eqeq1 2461 (in free logic, the same proof holds since one has equality axioms for terms). (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.)

 |-  ( E. x  x  =  A 
 <-> 
 E. y  y  =  A )
 
Theorembj-issetwt 34536* Closed form of bj-issetw 34537. (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.)
 |-  ( A. x ph  ->  ( A  e.  { x  |  ph }  <->  E. y  y  =  A ) )
 
Theorembj-issetw 34537* The closest one can get to isset 3113 without using ax-ext 2435. See also bj-vexw 34532. Note that the only dv condition is between  y and  A. (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.)
 |-  ph   =>    |-  ( A  e.  { x  |  ph }  <->  E. y  y  =  A )
 
Theorembj-elissetv 34538* Version of bj-elisset 34539 with a dv condition on  x ,  V. This proof uses only df-ex 1614, ax-gen 1619, ax-4 1632 and df-clel 2452 on top of propositional calculus. Prefer its use over bj-elisset 34539 when sufficient. (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)
 |-  ( A  e.  V  ->  E. x  x  =  A )
 
Theorembj-elisset 34539* Remove from elisset 3120 dependency on ax-ext 2435 (and on df-cleq 2449 and df-v 3111). This proof uses only df-clab 2443 and df-clel 2452 on top of first-order logic. It only uses ax-12 1855 among the auxiliary logical axioms. Use bj-elissetv 34538 instead when sufficient (in particular when  V is substituted for  _V). (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.)
 |-  ( A  e.  V  ->  E. x  x  =  A )
 
Theorembj-issetiv 34540* Version of bj-isseti 34541 with a dv condition on  x ,  V. This proof uses only df-ex 1614, ax-gen 1619, ax-4 1632 and df-clel 2452 on top of propositional calculus. Prefer its use over bj-isseti 34541 when sufficient (in particular when  V is substituted for  _V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)
 |-  A  e.  V   =>    |- 
 E. x  x  =  A
 
Theorembj-isseti 34541* Remove from isseti 3115 dependency on ax-ext 2435 (and on df-cleq 2449 and df-v 3111). This proof uses only df-clab 2443 and df-clel 2452 on top of first-order logic. It only uses ax-12 1855 among the auxiliary logical axioms. The hypothesis uses 
V instead of  _V for extra generality. This is indeed more general as long as elex 3118 is not available. Use bj-issetiv 34540 instead when sufficient (in particular when  V is substituted for  _V). (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.)
 |-  A  e.  V   =>    |- 
 E. x  x  =  A
 
Theorembj-ralvw 34542 A weak version of ralv 3123 not using ax-ext 2435 (nor df-cleq 2449, df-clel 2452, df-v 3111), but using ax-13 2000. For the sake of illustration, the next theorem bj-rexvwv 34543, a weak version of rexv 3124, has a dv condition and avoids dependency on ax-13 2000, while the analogues for reuv 3125 and rmov 3126 are not proved. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
 |-  ps   =>    |-  ( A. x  e.  { y  |  ps } ph  <->  A. x ph )
 
Theorembj-rexvwv 34543* A weak version of rexv 3124 not using ax-ext 2435 (nor df-cleq 2449, df-clel 2452, df-v 3111) with an additional dv condition to avoid dependency on ax-13 2000 as well. See bj-ralvw 34542. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
 |-  ps   =>    |-  ( E. x  e.  { y  |  ps } ph  <->  E. x ph )
 
Theorembj-rababwv 34544* A weak version of rabab 3127 not using df-clel 2452 nor df-v 3111 (but requiring ax-ext 2435). A version without dv condition is provable by replacing bj-vexwv 34534 with bj-vexw 34532 in the proof, hence requiring ax-13 2000. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
 |-  ps   =>    |-  { x  e.  { y  |  ps }  |  ph }  =  { x  |  ph }
 
Theorembj-ralcom4 34545* Remove from ralcom4 3128 dependency on ax-ext 2435 and ax-13 2000 (and on df-or 370, df-an 371, df-tru 1398, df-sb 1741, df-clab 2443, df-cleq 2449, df-clel 2452, df-nfc 2607, df-v 3111). This proof uses only df-ral 2812 on top of first-order logic. (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.)
 |-  ( A. x  e.  A  A. y ph  <->  A. y A. x  e.  A  ph )
 
Theorembj-rexcom4 34546* Remove from rexcom4 3129 dependency on ax-ext 2435 and ax-13 2000 (and on df-or 370, df-tru 1398, df-sb 1741, df-clab 2443, df-cleq 2449, df-clel 2452, df-nfc 2607, df-v 3111). This proof uses only df-rex 2813 on top of first-order logic. (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.)
 |-  ( E. x  e.  A  E. y ph  <->  E. y E. x  e.  A  ph )
 
Theorembj-rexcom4a 34547* Remove from rexcom4a 3130 dependency on ax-ext 2435 and ax-13 2000 (and on df-or 370, df-sb 1741, df-clab 2443, df-cleq 2449, df-clel 2452, df-nfc 2607, df-v 3111). This proof uses only df-rex 2813 on top of first-order logic. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
 |-  ( E. x E. y  e.  A  ( ph  /\  ps ) 
 <-> 
 E. y  e.  A  ( ph  /\  E. x ps ) )
 
Theorembj-rexcom4bv 34548* Version of bj-rexcom4b 34549 with a dv condition on  x ,  V, hence removing dependency on df-sb 1741 and df-clab 2443 (so that it depends on df-clel 2452 and df-rex 2813 only on top of first-order logic). Prefer its use over bj-rexcom4b 34549 when sufficient (in particular when  V is substituted for  _V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)
 |-  B  e.  V   =>    |-  ( E. x E. y  e.  A  ( ph  /\  x  =  B ) 
 <-> 
 E. y  e.  A  ph )
 
Theorembj-rexcom4b 34549* Remove from rexcom4b 3131 dependency on ax-ext 2435 and ax-13 2000 (and on df-or 370, df-cleq 2449, df-nfc 2607, df-v 3111). The hypothesis uses  V instead of  _V (see bj-isseti 34541 for the motivation). Use bj-rexcom4bv 34548 instead when sufficient (in particular when  V is substituted for  _V). (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
 |-  B  e.  V   =>    |-  ( E. x E. y  e.  A  ( ph  /\  x  =  B ) 
 <-> 
 E. y  e.  A  ph )
 
Theorembj-ceqsalt0 34550 The FOL content of ceqsalt 3132. Lemma for bj-ceqsalt 34552 and bj-ceqsaltv 34553. (Contributed by BJ, 26-Sep-2019.) (Proof modification is discouraged.)
 |-  (
 ( F/ x ps  /\ 
 A. x ( th  ->  ( ph  <->  ps ) )  /\  E. x th )  ->  ( A. x ( th  -> 
 ph )  <->  ps ) )
 
Theorembj-ceqsalt1 34551 The FOL content of ceqsalt 3132. Lemma for bj-ceqsalt 34552 and bj-ceqsaltv 34553. (TODO: consider removing if it does not add anything to bj-ceqsalt0 34550.) (Contributed by BJ, 26-Sep-2019.) (Proof modification is discouraged.)
 |-  ( th  ->  E. x ch )   =>    |-  (
 ( F/ x ps  /\ 
 A. x ( ch 
 ->  ( ph  <->  ps ) )  /\  th )  ->  ( A. x ( ch  ->  ph )  <->  ps ) )
 
Theorembj-ceqsalt 34552* Remove from ceqsalt 3132 dependency on ax-ext 2435 (and on df-cleq 2449 and df-v 3111). Note: this is not doable with ceqsralt 3133 (or ceqsralv 3138), which uses eleq1 2529, but the same dependence removal is possible for ceqsalg 3134, ceqsal 3136, ceqsalv 3137, cgsexg 3142, cgsex2g 3143, cgsex4g 3144, ceqsex 3145, ceqsexv 3146, ceqsex2 3147, ceqsex2v 3148, ceqsex3v 3149, ceqsex4v 3150, ceqsex6v 3151, ceqsex8v 3152, gencbvex 3153 (after changing  A  =  y to  y  =  A), gencbvex2 3154, gencbval 3155, vtoclgft 3157 (it uses  F/_, whose justification nfcjust 2606 is actually provable without ax-ext 2435, as bj-nfcjust 34527 shows) and several other vtocl* theorems (see for instance bj-vtoclg1f 34584). See also bj-ceqsaltv 34553. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
 |-  (
 ( F/ x ps  /\ 
 A. x ( x  =  A  ->  ( ph 
 <->  ps ) )  /\  A  e.  V )  ->  ( A. x ( x  =  A  ->  ph )  <->  ps ) )
 
Theorembj-ceqsaltv 34553* Version of bj-ceqsalt 34552 with a dv condition on  x ,  V, removing dependency on df-sb 1741 and df-clab 2443. Prefer its use over bj-ceqsalt 34552 when sufficient (in particular when  V is substituted for  _V). (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
 |-  (
 ( F/ x ps  /\ 
 A. x ( x  =  A  ->  ( ph 
 <->  ps ) )  /\  A  e.  V )  ->  ( A. x ( x  =  A  ->  ph )  <->  ps ) )
 
Theorembj-ceqsalg0 34554 The FOL content of ceqsalg 3134. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
 |-  F/ x ps   &    |-  ( ch  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x ch  ->  ( A. x ( ch  ->  ph )  <->  ps ) )
 
Theorembj-ceqsalg 34555* Remove from ceqsalg 3134 dependency on ax-ext 2435 (and on df-cleq 2449 and df-v 3111). See also bj-ceqsalgv 34557. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
 |-  F/ x ps   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  ph )  <->  ps ) )
 
Theorembj-ceqsalgALT 34556* Alternate proof of bj-ceqsalg 34555. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  F/ x ps   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  ph )  <->  ps ) )
 
Theorembj-ceqsalgv 34557* Version of bj-ceqsalg 34555 with a dv condition on  x ,  V, removing dependency on df-sb 1741 and df-clab 2443. Prefer its use over bj-ceqsalg 34555 when sufficient (in particular when  V is substituted for  _V). (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
 |-  F/ x ps   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  ph )  <->  ps ) )
 
Theorembj-ceqsalgvALT 34558* Alternate proof of bj-ceqsalgv 34557. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  F/ x ps   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  ph )  <->  ps ) )
 
Theorembj-ceqsal 34559* Remove from ceqsal 3136 dependency on ax-ext 2435 (and on df-cleq 2449, df-v 3111, df-clab 2443, df-sb 1741). (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
 |-  F/ x ps   &    |-  A  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ( x  =  A  -> 
 ph )  <->  ps )
 
Theorembj-ceqsalv 34560* Remove from ceqsalv 3137 dependency on ax-ext 2435 (and on df-cleq 2449, df-v 3111, df-clab 2443, df-sb 1741). (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A. x ( x  =  A  ->  ph )  <->  ps )
 
21.29.5.3  The class-form not-free predicate

In this section, we prove the symmetry of the class-form not-free predicate.

 
Theorembj-nfcsym 34561 The class-form not-free predicate defines a symmetric binary relation on var metavariables (irreflexivity is proved by nfnid 4685 with additional axioms; see also nfcv 2619). This could be proved from aecom 2052 and nfcvb 4686 but the latter requires a domain with at least two objects (hence uses extra axioms). (Contributed by BJ, 30-Sep-2018.) Proof modification is discouraged to avoid use of eqcomd 2465 instead of bj-equcomd 34335; removing dependency on ax-ext 2435 is possible: prove weak versions (i.e. replace classes with setvars) of drnfc1 2638, eleq2d 2527 (using elequ2 1824), nfcvf 2644, dvelimc 2643, dvelimdc 2642, nfcvf2 2645. (Proof modification is discouraged.)
 |-  ( F/_ x y  <->  F/_ y x )
 
21.29.5.4  Proposal for the definitions of class membership and class equality

In this section, we show (bj-ax8 34562 and bj-ax9 34565) that the current forms of the definitions of class membership (df-clel 2452) and class equality (df-cleq 2449) are too powerful, and we propose alternate definitions (bj-df-clel 34563 and bj-df-cleq 34566) which also have the advantage of making it clear that these definitions are conservative.

 
Theorembj-ax8 34562 Proof of ax-8 1821 from df-clel 2452 (and FOL). This shows that df-clel 2452 is "too powerful". A possible definition is given by bj-df-clel 34563. (Contributed by BJ, 27-Jun-2019.) (Proof modification is discouraged.)
 |-  ( x  =  y  ->  ( x  e.  z  ->  y  e.  z )
 )
 
Theorembj-df-clel 34563* Candidate definition for df-clel 2452 (the need for it is exposed in bj-ax8 34562). The similarity of the hypothesis and the conclusion makes it clear that this definition merely extends to class variables something that is true for setvar variables, hence is conservative. This definition should be directly referenced only by bj-dfclel 34564, which should be used instead. The proof is irrelevant since this is a proposal for an axiom.

Note: the current definition df-clel 2452 already mentions cleljust 2110 as a justification; here, we merely propose to put it as a hypothesis to make things clearer. (Contributed by BJ, 27-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

 |-  (
 y  e.  z  <->  E. x ( x  =  y  /\  x  e.  z ) )   =>    |-  ( A  e.  B 
 <-> 
 E. x ( x  =  A  /\  x  e.  B ) )
 
Theorembj-dfclel 34564* Characterization of the elements of a class. (Contributed by BJ, 27-Jun-2019.) (Proof modification is discouraged.)
 |-  ( A  e.  B  <->  E. x ( x  =  A  /\  x  e.  B ) )
 
Theorembj-ax9 34565* Proof of ax-9 1823 from ax-ext 2435 and df-cleq 2449 (and FOL). This shows that df-cleq 2449 is "too powerful". A possible definition is given by bj-df-cleq 34566. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
 |-  ( x  =  y  ->  ( z  e.  x  ->  z  e.  y )
 )
 
Theorembj-df-cleq 34566* Candidate definition for df-cleq 2449 (the need for it is exposed in bj-ax9 34565). The similarity of the hypothesis and the conclusion makes it clear that this definition merely extends to class variables something that is true for setvar variables, hence is conservative. This definition should be directly referenced only by bj-dfcleq 34567, which should be used instead. The proof is irrelevant since this is a proposal for an axiom.

Another definition, which would give finer control, is actually the pair of definitions where each has one implication of the present biconditional as hypothesis and conclusion. They assert that extensionality (respectively, the left-substitution axiom for the membership predicate) extends from setvars to classes. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

 |-  (
 y  =  z  <->  A. x ( x  e.  y  <->  x  e.  z
 ) )   =>    |-  ( A  =  B  <->  A. x ( x  e.  A  <->  x  e.  B ) )
 
Theorembj-dfcleq 34567* Proof of class extensionality from the axiom of set extensionality (ax-ext 2435) and the definition of class equality (bj-df-cleq 34566). (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
 |-  ( A  =  B  <->  A. x ( x  e.  A  <->  x  e.  B ) )
 
21.29.5.5  Lemmas for class substitution

Some useful theorems for dealing with substitutions: sbbi 2143, sbcbig 3374, sbcel1g 3837, sbcel2 3839, sbcel12 3832, sbceqg 3834, csbvarg 3855.

 
Theorembj-sbeqALT 34568* Substitution in an equality (use the more genereal version bj-sbeq 34569 instead, without disjoint variable condition). (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
 |-  ( [ y  /  x ] A  =  B  <->  [_ y  /  x ]_ A  =  [_ y  /  x ]_ B )
 
Theorembj-sbeq 34569 Distribute proper substitution through an equality relation. (See sbceqg 3834). (Contributed by BJ, 6-Oct-2018.)
 |-  ( [ y  /  x ] A  =  B  <->  [_ y  /  x ]_ A  =  [_ y  /  x ]_ B )
 
Theorembj-sbceqgALT 34570 Distribute proper substitution through an equality relation. Alternate proof of sbceqg 3834. (Contributed by BJ, 6-Oct-2018.) Proof modification is discouraged to avoid using sbceqg 3834, but "minimize */except sbceqg" is ok. (Proof modification is discouraged.)
 |-  ( A  e.  V  ->  (
 [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )
 
Theorembj-csbsnlem 34571* Lemma for bj-csbsn 34572 (in this lemma,  x cannot occur in  A). (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
 |-  [_ A  /  x ]_ { x }  =  { A }
 
Theorembj-csbsn 34572 Substitution in a singleton. (Contributed by BJ, 6-Oct-2018.)
 |-  [_ A  /  x ]_ { x }  =  { A }
 
Theorembj-sbel1 34573* Version of sbcel1g 3837 when substituting a set. (Note: one could have a corresponding version of sbcel12 3832 when substituting a set, but the point here is that the antecedent of sbcel1g 3837 is not needed when substituting a set.) (Contributed by BJ, 6-Oct-2018.)
 |-  ( [ y  /  x ] A  e.  B  <->  [_ y  /  x ]_ A  e.  B )
 
Theorembj-abtru 34574 The class of sets verifying a tautology is the universal class. (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
 |-  ( A. x ph  ->  { x  |  ph }  =  _V )
 
Theorembj-abfal 34575 The class of sets verifying a falsity is the empty set (closed form of abf 3828). (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
 |-  ( A. x  -.  ph  ->  { x  |  ph }  =  (/) )
 
Theorembj-abf 34576 Shorter proof of abf 3828 (which should be kept as abfALT). (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
 |-  -.  ph   =>    |-  { x  |  ph }  =  (/)
 
Theorembj-csbprc 34577 More direct proof of csbprc 3830 (fewer essential steps). (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
 |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ B  =  (/) )
 
21.29.5.6  Removing some dv conditions
 
Theorembj-exlimmpi 34578 Lemma for bj-vtoclg1f1 34583 (an instance of this lemma is a version of bj-vtoclg1f1 34583 where  x and  y are identified). (Contributed by BJ, 30-Apr-2019.) (Proof modification is discouraged.)
 |-  F/ x ps   &    |-  ( ch  ->  (
 ph  ->  ps ) )   &    |-  ph   =>    |-  ( E. x ch  ->  ps )
 
Theorembj-exlimmpbi 34579 Lemma for theorems of the vtoclg 3167 family. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.)
 |-  F/ x ps   &    |-  ( ch  ->  (
 ph 
 <->  ps ) )   &    |-  ph   =>    |-  ( E. x ch  ->  ps )
 
Theorembj-exlimmpbir 34580 Lemma for theorems of the vtoclg 3167 family. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.)
 |-  F/ x ph   &    |-  ( ch  ->  (
 ph 
 <->  ps ) )   &    |-  ps   =>    |-  ( E. x ch  ->  ph )
 
Theorembj-vtoclf 34581* Remove dependency on ax-ext 2435, df-clab 2443 and df-cleq 2449 (and df-sb 1741 and df-v 3111) from vtoclf 3160. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
 |-  F/ x ps   &    |-  A  e.  V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  ph   =>    |- 
 ps
 
Theorembj-vtocl 34582* Remove dependency on ax-ext 2435, df-clab 2443 and df-cleq 2449 (and df-sb 1741 and df-v 3111) from vtocl 3161. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
 |-  A  e.  V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ph   =>    |- 
 ps
 
Theorembj-vtoclg1f1 34583* The FOL content of vtoclg1f 3166 (hence not using ax-ext 2435, df-cleq 2449, df-nfc 2607, df-v 3111). Note the weakened "major" hypothesis and the dv condition between  x and  A (needed since the class-form non-free predicate is not available without ax-ext 2435; as a byproduct, this dispenses with ax-11 1843 and ax-13 2000). (Contributed by BJ, 30-Apr-2019.) (Proof modification is discouraged.)
 |-  F/ x ps   &    |-  ( x  =  A  ->  ( ph  ->  ps ) )   &    |-  ph   =>    |-  ( E. y  y  =  A  ->  ps )
 
Theorembj-vtoclg1f 34584* Reprove vtoclg1f 3166 from bj-vtoclg1f1 34583. This removes dependency on ax-ext 2435, df-cleq 2449 and df-v 3111. Use bj-vtoclg1fv 34585 instead when sufficient (in particular when  V is substituted for  _V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)
 |-  F/ x ps   &    |-  ( x  =  A  ->  ( ph  ->  ps ) )   &    |-  ph   =>    |-  ( A  e.  V  ->  ps )
 
Theorembj-vtoclg1fv 34585* Version of bj-vtoclg1f 34584 with a dv condition on  x ,  V. This removes dependency on df-sb 1741 and df-clab 2443. Prefer its use over bj-vtoclg1f 34584 when sufficient (in particular when  V is substituted for  _V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)
 |-  F/ x ps   &    |-  ( x  =  A  ->  ( ph  ->  ps ) )   &    |-  ph   =>    |-  ( A  e.  V  ->  ps )
 
Theorembj-rabbida2 34586 Version of rabbidva2 3099 with dv condition replaced by non-freeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
 |-  F/ x ph   &    |-  ( ph  ->  ( ( x  e.  A  /\  ps )  <->  ( x  e.  B  /\  ch )
 ) )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  =  { x  e.  B  |  ch } )
 
Theorembj-rabbida 34587 Version of rabbidva 3100 with dv condition replaced by non-freeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
 |-  F/ x ph   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  =  { x  e.  A  |  ch }
 )
 
Theorembj-rabbid 34588 Version of rabbidv 3101 with dv condition replaced by non-freeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
 |-  F/ x ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  =  { x  e.  A  |  ch }
 )
 
Theorembj-rabeqd 34589 Deduction form of rabeq 3103. Note that contrary to rabeq 3103 it has no dv condition. (Contributed by BJ, 27-Apr-2019.)
 |-  F/ x ph   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  =  { x  e.  B  |  ps }
 )
 
Theorembj-rabeqbid 34590 Version of rabeqbidv 3104 with two dv conditions removed and the third replaced by a non-freeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
 |-  F/ x ph   &    |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  =  { x  e.  B  |  ch }
 )
 
Theorembj-rabeqbida 34591 Version of rabeqbidva 3105 with two dv conditions removed and the third replaced by a non-freeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
 |-  F/ x ph   &    |-  ( ph  ->  A  =  B )   &    |-  (
 ( ph  /\  x  e.  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  =  { x  e.  B  |  ch }
 )
 
Theorembj-seex 34592* Version of seex 4851 with a dv condition replaced by a non-freeness hypothesis (for the sake of illustration). (Contributed by BJ, 27-Apr-2019.)
 |-  F/_ x B   =>    |-  ( ( R Se  A  /\  B  e.  A ) 
 ->  { x  e.  A  |  x R B }  e.  _V )
 
Theorembj-nfcf 34593* Version of df-nfc 2607 with a dv condition replaced with a non-freeness hypothesis. (Contributed by BJ, 2-May-2019.)
 |-  F/_ y A   =>    |-  ( F/_ x A  <->  A. y F/ x  y  e.  A )
 
Theorembj-axsep2 34594* Remove dependency on ax-13 2000, ax-ext 2435, df-cleq 2449 and df-clel 2452 from axsep2 4579 while shortening its proof. Remark: the comment in zfauscl 4580 is misleading: the essential use of ax-ext 2435 is the one via eleq2 2530 and not the one via vtocl 3161, since the latter can be proved without ax-ext 2435 (see bj-vtocl 34582). (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
 |-  E. y A. x ( x  e.  y  <->  ( x  e.  z  /\  ph )
 )
 
21.29.5.7  Class abstractions

A few additional theorems on class abstractions and restricted class abstractions.

 
Theorembj-unrab 34595* Generalization of unrab 3776. Equality need not hold. (Contributed by BJ, 21-Apr-2019.)
 |-  ( { x  e.  A  |  ph }  u.  { x  e.  B  |  ps } )  C_  { x  e.  ( A  u.  B )  |  ( ph  \/  ps ) }
 
Theorembj-inrab 34596 Generalization of inrab 3777. (Contributed by BJ, 21-Apr-2019.)
 |-  ( { x  e.  A  |  ph }  i^i  { x  e.  B  |  ps } )  =  { x  e.  ( A  i^i  B )  |  (
 ph  /\  ps ) }
 
Theorembj-inrab2 34597 Shorter proof of inrab 3777. (Contributed by BJ, 21-Apr-2019.) (Proof modification is discouraged.)
 |-  ( { x  e.  A  |  ph }  i^i  { x  e.  A  |  ps } )  =  { x  e.  A  |  ( ph  /\  ps ) }
 
Theorembj-rabtr 34598* Restricted class abstraction with true formula. (Contributed by BJ, 22-Apr-2019.)
 |-  { x  e.  A  | T.  }  =  A
 
Theorembj-rabtrALT 34599* Alternate proof of bj-rabtr 34598. (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  { x  e.  A  | T.  }  =  A
 
Theorembj-rabtrAUTO 34600* Proof of bj-rabtr 34598 found automatically by "improve all /depth 3 /3" followed by "minimize *". (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  { x  e.  A  | T.  }  =  A
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38122
  Copyright terms: Public domain < Previous  Next >