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Theorem List for Metamath Proof Explorer - 29401-29500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembnj1463 29401* Technical lemma for bnj60 29408. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  Q  e.  _V )   &    |-  ( ch  ->  A. z  e.  E  ( Q `  z )  =  ( G `  W ) )   &    |-  ( ch  ->  Q  Fn  E )   &    |-  ( ch  ->  E  e.  B )   =>    |-  ( ch  ->  Q  e.  C )
 
Theorembnj1489 29402* Technical lemma for bnj60 29408. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   =>    |-  ( ch  ->  Q  e.  _V )
 
Theorembnj1491 29403* Technical lemma for bnj60 29408. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  ( ch  ->  ( Q  e.  C  /\  dom 
 Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )   =>    |-  ( ( ch  /\  Q  e.  _V )  ->  E. f ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
 
Theorembnj1312 29404* Technical lemma for bnj60 29408. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   =>    |-  ( R  FrSe  A  ->  A. x  e.  A  E. f  e.  C  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
 
Theorembnj1493 29405* Technical lemma for bnj60 29408. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   =>    |-  ( R  FrSe  A  ->  A. x  e.  A  E. f  e.  C  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R )
 ) )
 
Theorembnj1497 29406* Technical lemma for bnj60 29408. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   =>    |-  A. g  e.  C  Fun  g
 
Theorembnj1498 29407* Technical lemma for bnj60 29408. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   =>    |-  ( R  FrSe  A  ->  dom 
 F  =  A )
 
18.26.5  Well-founded recursion, part 1 of 3
 
Theorembnj60 29408* Well-founded recursion, part 1 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   =>    |-  ( R  FrSe  A  ->  F  Fn  A )
 
Theorembnj1514 29409* Technical lemma for bnj1500 29414. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   =>    |-  (
 f  e.  C  ->  A. x  e.  dom  f
 ( f `  x )  =  ( G `  Y ) )
 
Theorembnj1518 29410* Technical lemma for bnj1500 29414. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   &    |-  ( ph  <->  ( R  FrSe  A 
 /\  x  e.  A ) )   &    |-  ( ps  <->  ( ph  /\  f  e.  C  /\  x  e. 
 dom  f ) )   =>    |-  ( ps  ->  A. d ps )
 
Theorembnj1519 29411* Technical lemma for bnj1500 29414. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   =>    |-  ( ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. )  ->  A. d
 ( F `  x )  =  ( G ` 
 <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. ) )
 
Theorembnj1520 29412* Technical lemma for bnj1500 29414. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   =>    |-  ( ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. )  ->  A. f
 ( F `  x )  =  ( G ` 
 <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. ) )
 
Theorembnj1501 29413* Technical lemma for bnj1500 29414. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   &    |-  ( ph  <->  ( R  FrSe  A 
 /\  x  e.  A ) )   &    |-  ( ps  <->  ( ph  /\  f  e.  C  /\  x  e. 
 dom  f ) )   &    |-  ( ch  <->  ( ps  /\  d  e.  B  /\  dom  f  =  d ) )   =>    |-  ( R  FrSe  A  ->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. ) )
 
18.26.6  Well-founded recursion, part 2 of 3
 
Theorembnj1500 29414* Well-founded recursion, part 2 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   =>    |-  ( R  FrSe  A  ->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. ) )
 
Theorembnj1525 29415* Technical lemma for bnj1522 29418. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   &    |-  ( ph  <->  ( R  FrSe  A 
 /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R )
 ) >. ) ) )   &    |-  ( ps  <->  ( ph  /\  F  =/=  H ) )   =>    |-  ( ps  ->  A. x ps )
 
Theorembnj1529 29416* Technical lemma for bnj1522 29418. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ch  ->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. ) )   &    |-  ( w  e.  F  ->  A. x  w  e.  F )   =>    |-  ( ch  ->  A. y  e.  A  ( F `  y )  =  ( G `  <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >. ) )
 
Theorembnj1523 29417* Technical lemma for bnj1522 29418. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   &    |-  ( ph  <->  ( R  FrSe  A 
 /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R )
 ) >. ) ) )   &    |-  ( ps  <->  ( ph  /\  F  =/=  H ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  A  /\  ( F `
  x )  =/=  ( H `  x ) ) )   &    |-  D  =  { x  e.  A  |  ( F `  x )  =/=  ( H `  x ) }   &    |-  ( th 
 <->  ( ch  /\  y  e.  D  /\  A. z  e.  D  -.  z R y ) )   =>    |-  ( ( R 
 FrSe  A  /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R )
 ) >. ) )  ->  F  =  H )
 
18.26.7  Well-founded recursion, part 3 of 3
 
Theorembnj1522 29418* Well-founded recursion, part 3 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   =>    |-  ( ( R  FrSe  A 
 /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R )
 ) >. ) )  ->  F  =  H )
 
18.27  Mathbox for Norm Megill

Note: A label suffixed with "N" (after the "Atoms..." section below), such as lshpnel2N 29797, means that the definition or theorem is not used for the derivation of hlathil 32776. This is a temporary renaming to assist cleaning up the theorems needed by hlathil 32776.

Please inform me of any changes that might affect my mathbox, since I may be working on it independently of the github commits. - Norm 30-Nov-2015

 
18.27.1  Experiments to study ax-7 unbundling

This section reproduces all predicate calculus theorems through sbal2 2086 that depend on ax-7 1720. It is an experiment to see how much of predicate calculus can be derived using the weaker (unbundled) ax-7v 29419.

The theorems in this section with suffix "NEW7" are direct replacements for the existing ones without the suffix but have proofs that avoid ax-7 1720 in favor of ax-7v 29419.

Theorems with suffix "AUX7" are new theorems that do not appear in the main predicate calculus section but assist the proofs of the "NEW7" suffixed theorems. They also use at most ax-7v 29419 and not ax-7 1720.

Theorems with suffix "OLD7" are the remaining predicate calculus theorems (through sbal2 2086) that haven't been proved from ax-7v 29419. In order to isolate them, they are derived from ax-7OLD7 29632 which replicates ax-7 1720. Whenever a proof of a *OLD7 theorem is found from ax-7v 29419, the suffix is changed to "NEW7" and the theorem is moved up to the "NEW7" section.

Theorems with suffix "AUXOLD7" (currently just nfsb4tw2AUXOLD7 29700) are results of an unsuccessful attempt to prove a helper theorem from ax-7v 29419, but still needs the help of ax-7 1720.

Currently there are about 137 "NEW7" theorems (starting after ax-7v 29419) and 91 "OLD7" theorems (starting after ax-7OLD7 29632).

 
18.27.1.1  Theorems derived from ax-7v (suffixes NEW7 and AUX7)
 
Axiomax-7v 29419* Experiment to see if ax-7 1720 can be unbundled i.e. can be derived from ax-7v 29419. This axiom is temporary. It will be replaced with a theorem derived from ax-7 1720 if we are successful, otherwise will be deleted. (Contributed by NM, 9-Oct-2017.)
 |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Theoremax7vAUX7 29420* A weaker version of ax-7 1720 with distinct variable restrictions. In order to show that this weakening is adequate, this should be the only theorem referencing ax-7 1720 directly.

(Right now we derive this from the temporary axiom ax-7v 29419 for easier 'show trace'. If this project is successful, which is seeming more and more unlikely, we will derive this from ax-7 1720 just as we derive ax9v 1645 from ax-9 1644.)

(Contributed by NM, 9-Oct-2017.)

 |-  ( A. x A. y ph  ->  A. y A. x ph )
 
TheoremalcomwAUX7 29421* Weak version of alcom 1723 not requiring ax-7 1720. (Contributed by NM, 27-Oct-2017.)
 |-  ( A. x A. y ph  <->  A. y A. x ph )
 
Theorema7swAUX7 29422* Weak version of a7s 1721 not requiring ax-7 1720. (Contributed by NM, 28-Oct-2017.)
 |-  ( A. x A. y ph  ->  ps )   =>    |-  ( A. y A. x ph  ->  ps )
 
Theoremcbv3hvNEW7 29423* Lemma for ax10NEW7 29450. Similar to cbv3h 1936. Requires distinct variables but avoids ax-12 1878. (Contributed by NM, 25-Jul-2015.) (Revised by NM, 25-Nov-2017.) Revised to prove from ax-7v 29419 instead of ax-7 1720.
 |-  ( ph  ->  A. y ph )   &    |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  A. y ps )
 
Theoremhbalw2AUX7 29424* Weak version of hbal 1722 not requiring ax-7 1720. (Contributed by NM, 9-Oct-2017.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( A. y ph  ->  A. x A. y ph )
 
TheoremhbaldwAUX7 29425* Weak version of hbald 1726 not requiring ax-7 1720. (Contributed by NM, 9-Oct-2017.)
 |-  ( ph  ->  A. y ph )   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   =>    |-  ( ph  ->  ( A. y ps  ->  A. x A. y ps ) )
 
TheoremhbexwAUX7 29426* Weak version of hbex 1745 not requiring ax-7 1720. (Contributed by NM, 9-Oct-2017.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( E. y ph  ->  A. x E. y ph )
 
TheoremnfalwAUX7 29427* Weak version of nfal 1778 not requiring ax-7 1720. (Contributed by NM, 27-Oct-2017.)
 |-  F/ x ph   =>    |- 
 F/ x A. y ph
 
TheoremnfexwAUX7 29428* Weak version of nfex 1779 not requiring ax-7 1720. (Contributed by NM, 27-Oct-2017.)
 |-  F/ x ph   =>    |- 
 F/ x E. y ph
 
TheoremnfaldwAUX7 29429* Weak version of nfald 1787 not requiring ax-7 1720. (Contributed by NM, 27-Oct-2017.)
 |-  F/ y ph   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x A. y ps )
 
TheoremnfexdwAUX7 29430* Weak version of nfexd 1788 not requiring ax-7 1720. (Contributed by NM, 27-Oct-2017.)
 |-  F/ y ph   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x E. y ps )
 
Theorem19.12vAUX7 29431* Weak version of 19.12 1746 not requiring ax-7 1720. (Contributed by NM, 10-Oct-2017.)
 |-  ( E. x A. y ph  ->  A. y E. x ph )
 
TheoremdvelimhwNEW7 29432* Proof of dvelimh 1917 without using ax-12 1878 but with additional distinct variable conditions. (Contributed by Andrew Salmon, 21-Jul-2011.) (Revised by NM, 1-Aug-2017.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. z ps )   &    |-  (
 z  =  y  ->  ( ph  <->  ps ) )   &    |-  ( -.  A. x  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
 )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
Theoremax12olem2wAUX7 29433* Lemma for ax12o 1887. Negate the equalities in ax-12 1878, shown as the hypothesis. (Contributed by NM, 10-Oct-2017.)
 |-  ( -.  x  =  y  ->  ( y  =  w 
 ->  A. x  y  =  w ) )   =>    |-  ( -.  x  =  y  ->  ( -.  y  =  z  ->  A. x  -.  y  =  z ) )
 
Theoremax12olem3aAUX7 29434 Lemma for ax12o 1887. Show the equivalence of an intermediate equivalent to ax12o 1887 with the conjunction of ax-12 1878 and a variant with negated equalities. (Contributed by NM, 29-Oct-2017.)
 |-  (
 ( ph  ->  ( -. 
 A. x  -.  ps  ->  A. y ps )
 ) 
 <->  ( ( ph  ->  ( ps  ->  A. y ps ) )  /\  ( ph  ->  ( -.  ps  ->  A. x  -.  ps ) ) ) )
 
Theoremax12olem4wAUX7 29435* Lemma for ax12o 1887. Construct an intermediate equivalent to ax-12 1878 from two instances of ax-12 1878. (Contributed by NM, 10-Oct-2017.)
 |-  ( -.  x  =  y  ->  ( y  =  z 
 ->  A. x  y  =  z ) )   &    |-  ( -.  x  =  y  ->  ( y  =  w 
 ->  A. x  y  =  w ) )   =>    |-  ( -.  x  =  y  ->  ( -. 
 A. x  -.  y  =  z  ->  A. x  y  =  z )
 )
 
Theoremax12olem6NEW7 29436* Lemma for ax12o 1887. Derivation of ax12o 1887 from the hypotheses, without using ax12o 1887. (Contributed by Andrew Salmon, 21-Jul-2011.) (Revised by NM, 24-Dec-2015.)
 |-  ( -.  A. x  x  =  z  ->  ( z  =  w  ->  A. x  z  =  w )
 )   &    |-  ( -.  A. x  x  =  y  ->  ( y  =  w  ->  A. x  y  =  w ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( -.  A. x  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z ) ) )
 
Theoremax12olem7NEW7 29437* Lemma for ax12o 1887. Derivation of ax12o 1887 from the hypotheses, without using ax12o 1887. (Contributed by NM, 24-Dec-2015.)
 |-  ( -.  x  =  z  ->  ( -.  A. x  -.  z  =  w  ->  A. x  z  =  w ) )   &    |-  ( -.  x  =  y  ->  ( -.  A. x  -.  y  =  w  ->  A. x  y  =  w ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( -.  A. x  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z ) ) )
 
Theoremax12oNEW7 29438 Derive set.mm's original ax-12o 2094 from the shorter ax-12 1878. (Contributed by NM, 29-Nov-2015.) (Revised by NM, 24-Dec-2015.)
 |-  ( -.  A. z  z  =  x  ->  ( -.  A. z  z  =  y 
 ->  ( x  =  y 
 ->  A. z  x  =  y ) ) )
 
TheoremdvelimvNEW7 29439* Similar to dvelim 1969 with first hypothesis replaced by distinct variable condition. (Contributed by NM, 25-Jul-2015.)
 |-  (
 z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
Theoremdveeq2NEW7 29440* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) (Revised by NM, 20-Jul-2015.)
 |-  ( -.  A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
 )
 
Theoremdveeq1NEW7 29441* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
 |-  ( -.  A. x  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
 )
 
Theoremdveel1NEW7 29442* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
 |-  ( -.  A. x  x  =  y  ->  ( y  e.  z  ->  A. x  y  e.  z )
 )
 
Theoremdveel2NEW7 29443* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
 |-  ( -.  A. x  x  =  y  ->  ( z  e.  y  ->  A. x  z  e.  y )
 )
 
TheoremdvelimwAUX7 29444* Weaker version of dvelim 1969. (Contributed by NM, 23-Nov-1994.)
 |-  ( ph  ->  A. x ph )   &    |-  (
 z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
Theoremax9NEW7 29445 Theorem showing that ax-9 1644 follows from the weaker version ax9v 1645. (Even though this theorem depends on ax-9 1644, all references of ax-9 1644 are made via ax9v 1645. An earlier version stated ax9v 1645 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-9 1644 so that all proofs can be traced back to ax9v 1645. (Contributed by NM, 12-Nov-2013.) (Revised by NM, 25-Jul-2015.)

 |-  -.  A. x  -.  x  =  y
 
Theoremax9oNEW7 29446 Show that the original axiom ax-9o 2090 can be derived from ax9 1902 and others. See ax9from9o 2100 for the rederivation of ax9 1902 from ax-9o 2090.

Normally, ax9o 1903 should be used rather than ax-9o 2090, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.)

 |-  ( A. x ( x  =  y  ->  A. x ph )  ->  ph )
 
Theorema9eNEW7 29447 At least one individual exists. This is not a theorem of free logic, which is sound in empty domains. For such a logic, we would add this theorem as an axiom of set theory (Axiom 0 of [Kunen] p. 10). In the system consisting of ax-5 1547 through ax-14 1700 and ax-17 1606, all axioms other than ax9 1902 are believed to be theorems of free logic, although the system without ax9 1902 is probably not complete in free logic. (Contributed by NM, 5-Aug-1993.)
 |-  E. x  x  =  y
 
Theoremax10lem4NEW7 29448* Lemma for ax10 1897. Change bound variable. (Contributed by NM, 8-Jul-2016.)
 |-  ( A. x  x  =  w  ->  A. y  y  =  x )
 
Theoremax10lem5NEW7 29449* Lemma for ax10 1897. Change free and bound variables. (Contributed by NM, 22-Jul-2015.)
 |-  ( A. z  z  =  w  ->  A. y  y  =  x )
 
Theoremax10NEW7 29450 Derive set.mm's original ax-10 2092 from others. (Contributed by NM, 25-Jul-2015.) (Revised by NM, 7-Nov-2015.)
 |-  ( A. x  x  =  y  ->  A. y  y  =  x )
 
TheoremaecomNEW7 29451 Commutation law for identical variable specifiers. The antecedent and consequent are true when  x and  y are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x  x  =  y  ->  A. y  y  =  x )
 
TheoremaecomsNEW7 29452 A commutation rule for identical variable specifiers. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x  x  =  y  ->  ph )   =>    |-  ( A. y  y  =  x  ->  ph )
 
Theoremax10oNEW7 29453 Show that ax-10o 2091 can be derived from ax-10 2092 in the form of ax10 1897. Normally, ax10o 1905 should be used rather than ax-10o 2091, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  y  ->  ( A. x ph 
 ->  A. y ph )
 )
 
Theoremhba1eAUX7 29454 Special case of hbae 1906 not requiring ax-7 1720. (Contributed by NM, 12-Oct-2017.)
 |-  ( A. x  x  =  y  ->  A. y A. x  x  =  y )
 
TheoremhbaewAUX7 29455* Weak version of hbae 1906 not requiring ax-7 1720. See hbaew2AUX7 29456 and hbaew3AUX7 29499 for versions with different distinct variable requirements. (Contributed by NM, 10-Oct-2017.)
 |-  ( A. x  x  =  y  ->  A. z A. x  x  =  y )
 
Theoremhbaew2AUX7 29456* Weak version of hbae 1906 not requiring ax-7 1720. Different distinct variable requirements from hbaewAUX7 29455. (Contributed by NM, 30-Oct-2017.)
 |-  ( A. x  x  =  y  ->  A. z A. x  x  =  y )
 
TheoremnfaewAUX7 29457* Weak version of nfae 1907 not requiring ax-7 1720. (Contributed by NM, 10-Oct-2017.)
 |-  F/ z A. x  x  =  y
 
TheoremhbnaewAUX7 29458* Weak version of hbnae 1908 not requiring ax-7 1720. (Contributed by NM, 10-Oct-2017.)
 |-  ( -.  A. x  x  =  y  ->  A. z  -.  A. x  x  =  y )
 
TheoremnfnaewAUX7 29459* Weak version of nfnae 1909 not requiring ax-7 1720. (Contributed by NM, 27-Oct-2017.)
 |-  F/ z  -.  A. x  x  =  y
 
Theoremnfaew2AUX7 29460* Weak version of nfae 1907 not requiring ax-7 1720. (Contributed by NM, 25-Nov-2017.)
 |-  F/ z A. x  x  =  y
 
Theoremhbnaew2AUX7 29461* Weak version of hbnae 1908 not requiring ax-7 1720. (Contributed by NM, 25-Nov-2017.)
 |-  ( -.  A. x  x  =  y  ->  A. z  -.  A. x  x  =  y )
 
Theoremnfnaew2AUX7 29462* Weak version of nfnae 1909 not requiring ax-7 1720. (Contributed by NM, 25-Nov-2017.)
 |-  F/ z  -.  A. x  x  =  y
 
TheoremnfeqfNEW7 29463 A variable is effectively not free in an equality if it is not either of the involved variables.  F/ version of ax-12o 2094. (Contributed by Mario Carneiro, 6-Oct-2016.)
 |-  (
 ( -.  A. z  z  =  x  /\  -. 
 A. z  z  =  y )  ->  F/ z  x  =  y )
 
TheoremequsalNEW7 29464 A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.)
 |-  F/ x ps   &    |-  ( x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x ( x  =  y  ->  ph )  <->  ps )
 
TheoremequsalihAUX7 29465 One direction of equsalhNEW7 29466 with weaker hypothesis. TO DO: Delete if not used. (Contributed by NM, 13-Nov-2017.)
 |-  ( x  =  y  ->  (
 ph  ->  A. x ps )
 )   =>    |-  ( A. x ( x  =  y  ->  ph )  ->  ps )
 
TheoremequsalhNEW7 29466 A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ( x  =  y  -> 
 ph )  <->  ps )
 
TheoremequsexNEW7 29467 A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)
 |-  F/ x ps   &    |-  ( x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( E. x ( x  =  y  /\  ph )  <->  ps )
 
TheoremequsexhNEW7 29468 A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x ( x  =  y  /\  ph )  <->  ps )
 
TheoremdvelimhvAUX7 29469* Weak version of dvelimh 1917 not requiring ax-7 1720. (Contributed by NM, 10-Oct-2017.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. z ps )   &    |-  (
 z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
Theoremdral1NEW7 29470 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 24-Nov-1994.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  (
 A. x ph  <->  A. y ps )
 )
 
Theoremdrex1NEW7 29471 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( E. x ph  <->  E. y ps )
 )
 
Theoremdrnf1NEW7 29472 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( F/ x ph  <->  F/ y ps )
 )
 
Theoremdral2wAUX7 29473* Weak version of dral2 1919 not requiring ax-7 1720. (Contributed by NM, 27-Oct-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  (
 A. z ph  <->  A. z ps )
 )
 
Theoremdrex2wAUX7 29474* Weak version of drex2 1921 not requiring ax-7 1720. (Contributed by NM, 27-Oct-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( E. z ph  <->  E. z ps )
 )
 
Theoremdrnf2wAUX7 29475* Weak version of drnf2 1923 not requiring ax-7 1720. (Contributed by NM, 27-Oct-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( F/ z ph  <->  F/ z ps )
 )
 
Theoremdral2w2AUX7 29476* Weak version of dral2 1919 not requiring ax-7 1720. (Contributed by NM, 25-Nov-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  (
 A. z ph  <->  A. z ps )
 )
 
Theoremdrex2w2AUX7 29477* Weak version of drex2 1921 not requiring ax-7 1720. (Contributed by NM, 25-Nov-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( E. z ph  <->  E. z ps )
 )
 
Theoremdrnf2w2AUX7 29478* Weak version of drnf2 1923 not requiring ax-7 1720. (Contributed by NM, 25-Nov-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( F/ z ph  <->  F/ z ps )
 )
 
Theoremdral2w3AUX7 29479 Weak version of dral2 1919 not requiring ax-7 1720. (Contributed by NM, 25-Nov-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  (
 A. x ph  <->  A. x ps )
 )
 
Theoremdrex2w3AUX7 29480 Weak version of drex2 1921 not requiring ax-7 1720. (Contributed by NM, 25-Nov-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( E. x ph  <->  E. x ps )
 )
 
Theoremdrnf2w3AUX7 29481 Weak version of drnf2 1923 not requiring ax-7 1720. (Contributed by NM, 25-Nov-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( F/ x ph  <->  F/ x ps )
 )
 
TheoremexdistrfNEW7 29482 Distribution of existential quantifiers, with a bound-variable hypothesis saying that  y is not free in  ph, but  x can be free in  ph (and there is no distinct variable condition on  x and  y). (Contributed by Mario Carneiro, 20-Mar-2013.) (Revised by NM, 25-Nov-2017.) Revised to prove from ax-7v 29419 instead of ax-7 1720.
 |-  ( -.  A. x  x  =  y  ->  F/ y ph )   =>    |-  ( E. x E. y ( ph  /\  ps )  ->  E. x ( ph  /\ 
 E. y ps )
 )
 
Theoremdrsb1NEW7 29483 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x  x  =  y  ->  ( [ z  /  x ] ph  <->  [ z  /  y ] ph ) )
 
TheoremspimtNEW7 29484 Closed theorem form of spim 1928. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.)
 |-  (
 ( F/ x ps  /\ 
 A. x ( x  =  y  ->  ( ph  ->  ps ) ) ) 
 ->  ( A. x ph  ->  ps ) )
 
TheoremspimNEW7 29485 Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1928 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)
 |-  F/ x ps   &    |-  ( x  =  y  ->  ( ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  ps )
 
TheoremspimeNEW7 29486 Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 3-Oct-2016.)
 |-  F/ x ph   &    |-  ( x  =  y  ->  ( ph  ->  ps ) )   =>    |-  ( ph  ->  E. x ps )
 
TheoremspimedNEW7 29487 Deduction version of spime 1929. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)
 |-  ( ch  ->  F/ x ph )   &    |-  ( x  =  y 
 ->  ( ph  ->  ps )
 )   =>    |-  ( ch  ->  ( ph  ->  E. x ps )
 )
 
Theoremcbv1hwAUX7 29488* Weak version of cbv1h 1931 not requiring ax-7 1720. (Contributed by NM, 28-Oct-2017.)
 |-  ( ph  ->  ( ps  ->  A. y ps ) )   &    |-  ( ph  ->  ( ch  ->  A. x ch )
 )   &    |-  ( ph  ->  ( x  =  y  ->  ( ps  ->  ch )
 ) )   =>    |-  ( A. x A. y ph  ->  ( A. x ps  ->  A. y ch ) )
 
Theoremcbv1wAUX7 29489* Weak version of cbv1 1932 not requiring ax-7 1720. (Contributed by NM, 28-Oct-2017.)
 |-  ( ph  ->  F/ y ps )   &    |-  ( ph  ->  F/ x ch )   &    |-  ( ph  ->  ( x  =  y  ->  ( ps  ->  ch ) ) )   =>    |-  ( A. x A. y ph  ->  ( A. x ps  ->  A. y ch )
 )
 
Theoremcbv2hwAUX7 29490* Weak version of cbv2h 1933 not requiring ax-7 1720. (Contributed by NM, 28-Oct-2017.)
 |-  ( ph  ->  ( ps  ->  A. y ps ) )   &    |-  ( ph  ->  ( ch  ->  A. x ch )
 )   &    |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch ) ) )   =>    |-  ( A. x A. y ph  ->  ( A. x ps 
 <-> 
 A. y ch )
 )
 
Theoremcbv2wAUX7 29491* Weak version of cbv2 1934 not requiring ax-7 1720. (Contributed by NM, 28-Oct-2017.)
 |-  ( ph  ->  F/ y ps )   &    |-  ( ph  ->  F/ x ch )   &    |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch ) ) )   =>    |-  ( A. x A. y ph  ->  ( A. x ps  <->  A. y ch )
 )
 
Theoremcbv3wAUX7 29492* Weak version of cbv3 1935 not requiring ax-7 1720. (Contributed by NM, 28-Oct-2017.)
 |-  F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  A. y ps )
 
Theoremcbv3hwAUX7 29493* Weak version of cbv3h 1936 not requiring ax-7 1720. (Contributed by NM, 28-Oct-2017.)
 |-  ( ph  ->  A. y ph )   &    |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  A. y ps )
 
TheoremcbvalwwAUX7 29494* Weak version of cbval 1937 not requiring ax-7 1720. (Contributed by NM, 28-Oct-2017.)
 |-  F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ph  <->  A. y ps )
 
TheoremcbvexwAUX7 29495* Weak version of cbvex 1938 not requiring ax-7 1720. (Contributed by NM, 28-Oct-2017.)
 |-  F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x ph  <->  E. y ps )
 
TheoremspimvNEW7 29496* A version of spim 1928 with a distinct variable requirement instead of a bound variable hypothesis. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  ps )
 
TheoremaevwAUX7 29497* Weak version of aev 1944 not requiring ax-7 1720. (Contributed by NM, 28-Oct-2017.)
 |-  ( A. x  x  =  y  ->  A. z  w  =  v )
 
TheoremaevNEW7 29498* A "distinctor elimination" lemma with no restrictions on variables in the consequent. (Contributed by NM, 8-Nov-2006.)
 |-  ( A. x  x  =  y  ->  A. z  w  =  v )
 
Theoremhbaew3AUX7 29499* Weak version of hbae 1906 not requiring ax-7 1720. Has different distinct variable requirements from hbaewAUX7 29455. (Contributed by NM, 30-Oct-2017.)
 |-  ( A. x  x  =  y  ->  A. z A. x  x  =  y )
 
Theoremnfaew3AUX7 29500* Weak version of nfae 1907 not requiring ax-7 1720. (Contributed by NM, 25-Nov-2017.)
 |-  F/ z A. x  x  =  y
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