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Theorem List for Intuitionistic Logic Explorer - 2601-2700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremceqsalv 2601* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ( x  =  A  -> 
 ph )  <->  ps )
 
Theoremceqsralv 2602* Restricted quantifier version of ceqsalv 2601. (Contributed by NM, 21-Jun-2013.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  B  ->  ( A. x  e.  B  ( x  =  A  ->  ph )  <->  ps ) )
 
Theoremgencl 2603* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
 |-  ( th  <->  E. x ( ch 
 /\  A  =  B ) )   &    |-  ( A  =  B  ->  ( ph  <->  ps ) )   &    |-  ( ch  ->  ph )   =>    |-  ( th  ->  ps )
 
Theorem2gencl 2604* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
 |-  ( C  e.  S  <->  E. x  e.  R  A  =  C )   &    |-  ( D  e.  S 
 <-> 
 E. y  e.  R  B  =  D )   &    |-  ( A  =  C  ->  (
 ph 
 <->  ps ) )   &    |-  ( B  =  D  ->  ( ps  <->  ch ) )   &    |-  (
 ( x  e.  R  /\  y  e.  R )  ->  ph )   =>    |-  ( ( C  e.  S  /\  D  e.  S )  ->  ch )
 
Theorem3gencl 2605* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
 |-  ( D  e.  S  <->  E. x  e.  R  A  =  D )   &    |-  ( F  e.  S 
 <-> 
 E. y  e.  R  B  =  F )   &    |-  ( G  e.  S  <->  E. z  e.  R  C  =  G )   &    |-  ( A  =  D  ->  (
 ph 
 <->  ps ) )   &    |-  ( B  =  F  ->  ( ps  <->  ch ) )   &    |-  ( C  =  G  ->  ( ch  <->  th ) )   &    |-  (
 ( x  e.  R  /\  y  e.  R  /\  z  e.  R )  ->  ph )   =>    |-  ( ( D  e.  S  /\  F  e.  S  /\  G  e.  S ) 
 ->  th )
 
Theoremcgsexg 2606* Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.)
 |-  ( x  =  A  ->  ch )   &    |-  ( ch  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A  e.  V  ->  ( E. x ( ch  /\  ph )  <->  ps ) )
 
Theoremcgsex2g 2607* Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.)
 |-  ( ( x  =  A  /\  y  =  B )  ->  ch )   &    |-  ( ch  ->  ( ph  <->  ps ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x E. y ( ch  /\  ph )  <->  ps ) )
 
Theoremcgsex4g 2608* An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.)
 |-  ( ( ( x  =  A  /\  y  =  B )  /\  (
 z  =  C  /\  w  =  D )
 )  ->  ch )   &    |-  ( ch  ->  ( ph  <->  ps ) )   =>    |-  ( ( ( A  e.  R  /\  B  e.  S )  /\  ( C  e.  R  /\  D  e.  S ) )  ->  ( E. x E. y E. z E. w ( ch  /\  ph )  <->  ps ) )
 
Theoremceqsex 2609* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.)
 |- 
 F/ x ps   &    |-  A  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( E. x ( x  =  A  /\  ph )  <->  ps )
 
Theoremceqsexv 2610* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x ( x  =  A  /\  ph )  <->  ps )
 
Theoremceqsex2 2611* Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)
 |- 
 F/ x ps   &    |-  F/ y ch   &    |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( y  =  B  ->  ( ps  <->  ch ) )   =>    |-  ( E. x E. y ( x  =  A  /\  y  =  B  /\  ph )  <->  ch )
 
Theoremceqsex2v 2612* Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( y  =  B  ->  ( ps  <->  ch ) )   =>    |-  ( E. x E. y ( x  =  A  /\  y  =  B  /\  ph )  <->  ch )
 
Theoremceqsex3v 2613* Elimination of three existential quantifiers, using implicit substitution. (Contributed by NM, 16-Aug-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 z  =  C  ->  ( ch  <->  th ) )   =>    |-  ( E. x E. y E. z ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ph )  <->  th )
 
Theoremceqsex4v 2614* Elimination of four existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( y  =  B  ->  ( ps  <->  ch ) )   &    |-  ( z  =  C  ->  ( ch  <->  th ) )   &    |-  ( w  =  D  ->  ( th  <->  ta ) )   =>    |-  ( E. x E. y E. z E. w ( ( x  =  A  /\  y  =  B )  /\  (
 z  =  C  /\  w  =  D )  /\  ph )  <->  ta )
 
Theoremceqsex6v 2615* Elimination of six existential quantifiers, using implicit substitution. (Contributed by NM, 21-Sep-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   &    |-  E  e.  _V   &    |-  F  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( y  =  B  ->  ( ps  <->  ch ) )   &    |-  ( z  =  C  ->  ( ch  <->  th ) )   &    |-  ( w  =  D  ->  ( th  <->  ta ) )   &    |-  ( v  =  E  ->  ( ta  <->  et ) )   &    |-  ( u  =  F  ->  ( et  <->  ze ) )   =>    |-  ( E. x E. y E. z E. w E. v E. u ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ( w  =  D  /\  v  =  E  /\  u  =  F )  /\  ph )  <->  ze )
 
Theoremceqsex8v 2616* Elimination of eight existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   &    |-  E  e.  _V   &    |-  F  e.  _V   &    |-  G  e.  _V   &    |-  H  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( y  =  B  ->  ( ps  <->  ch ) )   &    |-  ( z  =  C  ->  ( ch  <->  th ) )   &    |-  ( w  =  D  ->  ( th  <->  ta ) )   &    |-  ( v  =  E  ->  ( ta  <->  et ) )   &    |-  ( u  =  F  ->  ( et  <->  ze ) )   &    |-  ( t  =  G  ->  ( ze  <->  si ) )   &    |-  ( s  =  H  ->  ( si  <->  rh ) )   =>    |-  ( E. x E. y E. z E. w E. v E. u E. t E. s ( ( ( x  =  A  /\  y  =  B )  /\  ( z  =  C  /\  w  =  D ) )  /\  ( ( v  =  E  /\  u  =  F )  /\  (
 t  =  G  /\  s  =  H )
 )  /\  ph )  <->  rh )
 
Theoremgencbvex 2617* Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 |-  A  e.  _V   &    |-  ( A  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ( A  =  y  ->  ( ch  <->  th ) )   &    |-  ( th 
 <-> 
 E. x ( ch 
 /\  A  =  y ) )   =>    |-  ( E. x ( ch  /\  ph )  <->  E. y ( th  /\  ps ) )
 
Theoremgencbvex2 2618* Restatement of gencbvex 2617 with weaker hypotheses. (Contributed by Jeff Hankins, 6-Dec-2006.)
 |-  A  e.  _V   &    |-  ( A  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ( A  =  y  ->  ( ch  <->  th ) )   &    |-  ( th  ->  E. x ( ch 
 /\  A  =  y ) )   =>    |-  ( E. x ( ch  /\  ph )  <->  E. y ( th  /\  ps ) )
 
Theoremgencbval 2619* Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof rewritten by Jim Kingdon, 20-Jun-2018.)
 |-  A  e.  _V   &    |-  ( A  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ( A  =  y  ->  ( ch  <->  th ) )   &    |-  ( th 
 <-> 
 E. x ( ch 
 /\  A  =  y ) )   =>    |-  ( A. x ( ch  ->  ph )  <->  A. y ( th  ->  ps ) )
 
Theoremsbhypf 2620* Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf . (Contributed by Raph Levien, 10-Apr-2004.)
 |- 
 F/ x ps   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( y  =  A  ->  ( [
 y  /  x ] ph 
 <->  ps ) )
 
Theoremvtoclgft 2621 Closed theorem form of vtoclgf 2629. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.)
 |-  ( ( ( F/_ x A  /\  F/ x ps )  /\  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A. x ph )  /\  A  e.  V )  ->  ps )
 
Theoremvtocldf 2622 Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  ( ps  <->  ch ) )   &    |-  ( ph  ->  ps )   &    |-  F/ x ph   &    |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/ x ch )   =>    |-  ( ph  ->  ch )
 
Theoremvtocld 2623* Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  ( ps  <->  ch ) )   &    |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ch )
 
Theoremvtoclf 2624* Implicit substitution of a class for a setvar variable. This is a generalization of chvar 1656. (Contributed by NM, 30-Aug-1993.)
 |- 
 F/ x ps   &    |-  A  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ph   =>    |- 
 ps
 
Theoremvtocl 2625* Implicit substitution of a class for a setvar variable. (Contributed by NM, 30-Aug-1993.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  ph   =>    |- 
 ps
 
Theoremvtocl2 2626* Implicit substitution of classes for setvar variables. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   &    |-  ph   =>    |- 
 ps
 
Theoremvtocl3 2627* Implicit substitution of classes for setvar variables. (Contributed by NM, 3-Jun-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  (
 ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph  <->  ps ) )   &    |-  ph   =>    |- 
 ps
 
Theoremvtoclb 2628* Implicit substitution of a class for a setvar variable. (Contributed by NM, 23-Dec-1993.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  A  ->  ( ps  <->  th ) )   &    |-  ( ph 
 <->  ps )   =>    |-  ( ch  <->  th )
 
Theoremvtoclgf 2629 Implicit substitution of a class for a setvar variable, with bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Proof shortened by Mario Carneiro, 10-Oct-2016.)
 |-  F/_ x A   &    |-  F/ x ps   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  ph   =>    |-  ( A  e.  V  ->  ps )
 
Theoremvtoclg 2630* Implicit substitution of a class expression for a setvar variable. (Contributed by NM, 17-Apr-1995.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ph   =>    |-  ( A  e.  V  ->  ps )
 
Theoremvtoclbg 2631* Implicit substitution of a class for a setvar variable. (Contributed by NM, 29-Apr-1994.)
 |-  ( x  =  A  ->  ( ph  <->  ch ) )   &    |-  ( x  =  A  ->  ( ps  <->  th ) )   &    |-  ( ph 
 <->  ps )   =>    |-  ( A  e.  V  ->  ( ch  <->  th ) )
 
Theoremvtocl2gf 2632 Implicit substitution of a class for a setvar variable. (Contributed by NM, 25-Apr-1995.)
 |-  F/_ x A   &    |-  F/_ y A   &    |-  F/_ y B   &    |- 
 F/ x ps   &    |-  F/ y ch   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( y  =  B  ->  ( ps  <->  ch ) )   &    |-  ph   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ch )
 
Theoremvtocl3gf 2633 Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ y A   &    |-  F/_ z A   &    |-  F/_ y B   &    |-  F/_ z B   &    |-  F/_ z C   &    |- 
 F/ x ps   &    |-  F/ y ch   &    |-  F/ z th   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 z  =  C  ->  ( ch  <->  th ) )   &    |-  ph   =>    |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X ) 
 ->  th )
 
Theoremvtocl2g 2634* Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 25-Apr-1995.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   &    |-  ph   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ch )
 
Theoremvtoclgaf 2635* Implicit substitution of a class for a setvar variable. (Contributed by NM, 17-Feb-2006.) (Revised by Mario Carneiro, 10-Oct-2016.)
 |-  F/_ x A   &    |-  F/ x ps   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  ( x  e.  B  ->  ph )   =>    |-  ( A  e.  B  ->  ps )
 
Theoremvtoclga 2636* Implicit substitution of a class for a setvar variable. (Contributed by NM, 20-Aug-1995.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  e.  B  ->  ph )   =>    |-  ( A  e.  B  ->  ps )
 
Theoremvtocl2gaf 2637* Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 10-Aug-2013.)
 |-  F/_ x A   &    |-  F/_ y A   &    |-  F/_ y B   &    |- 
 F/ x ps   &    |-  F/ y ch   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( y  =  B  ->  ( ps  <->  ch ) )   &    |-  ( ( x  e.  C  /\  y  e.  D )  ->  ph )   =>    |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ch )
 
Theoremvtocl2ga 2638* Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 20-Aug-1995.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 ( x  e.  C  /\  y  e.  D )  ->  ph )   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ch )
 
Theoremvtocl3gaf 2639* Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 11-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ y A   &    |-  F/_ z A   &    |-  F/_ y B   &    |-  F/_ z B   &    |-  F/_ z C   &    |- 
 F/ x ps   &    |-  F/ y ch   &    |-  F/ z th   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 z  =  C  ->  ( ch  <->  th ) )   &    |-  (
 ( x  e.  R  /\  y  e.  S  /\  z  e.  T )  ->  ph )   =>    |-  ( ( A  e.  R  /\  B  e.  S  /\  C  e.  T ) 
 ->  th )
 
Theoremvtocl3ga 2640* Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 z  =  C  ->  ( ch  <->  th ) )   &    |-  (
 ( x  e.  D  /\  y  e.  R  /\  z  e.  S )  ->  ph )   =>    |-  ( ( A  e.  D  /\  B  e.  R  /\  C  e.  S ) 
 ->  th )
 
Theoremvtocleg 2641* Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Jan-2004.)
 |-  ( x  =  A  -> 
 ph )   =>    |-  ( A  e.  V  -> 
 ph )
 
Theoremvtoclegft 2642* Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 2643.) (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)
 |-  ( ( A  e.  B  /\  F/ x ph  /\ 
 A. x ( x  =  A  ->  ph )
 )  ->  ph )
 
Theoremvtoclef 2643* Implicit substitution of a class for a setvar variable. (Contributed by NM, 18-Aug-1993.)
 |- 
 F/ x ph   &    |-  A  e.  _V   &    |-  ( x  =  A  ->  ph )   =>    |-  ph
 
Theoremvtocle 2644* Implicit substitution of a class for a setvar variable. (Contributed by NM, 9-Sep-1993.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  ph )   =>    |-  ph
 
Theoremvtoclri 2645* Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Nov-1994.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  A. x  e.  B  ph   =>    |-  ( A  e.  B  ->  ps )
 
Theoremspcimgft 2646 A closed version of spcimgf 2650. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |- 
 F/ x ps   &    |-  F/_ x A   =>    |-  ( A. x ( x  =  A  ->  (
 ph  ->  ps ) )  ->  ( A  e.  B  ->  ( A. x ph  ->  ps ) ) )
 
Theoremspcgft 2647 A closed version of spcgf 2652. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 4-Jan-2017.)
 |- 
 F/ x ps   &    |-  F/_ x A   =>    |-  ( A. x ( x  =  A  ->  (
 ph 
 <->  ps ) )  ->  ( A  e.  B  ->  ( A. x ph  ->  ps ) ) )
 
Theoremspcimegft 2648 A closed version of spcimegf 2651. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |- 
 F/ x ps   &    |-  F/_ x A   =>    |-  ( A. x ( x  =  A  ->  ( ps  ->  ph ) ) 
 ->  ( A  e.  B  ->  ( ps  ->  E. x ph ) ) )
 
Theoremspcegft 2649 A closed version of spcegf 2653. (Contributed by Jim Kingdon, 22-Jun-2018.)
 |- 
 F/ x ps   &    |-  F/_ x A   =>    |-  ( A. x ( x  =  A  ->  (
 ph 
 <->  ps ) )  ->  ( A  e.  B  ->  ( ps  ->  E. x ph ) ) )
 
Theoremspcimgf 2650 Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  F/_ x A   &    |-  F/ x ps   &    |-  ( x  =  A  ->  (
 ph  ->  ps ) )   =>    |-  ( A  e.  V  ->  ( A. x ph 
 ->  ps ) )
 
Theoremspcimegf 2651 Existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  F/_ x A   &    |-  F/ x ps   &    |-  ( x  =  A  ->  ( ps  ->  ph ) )   =>    |-  ( A  e.  V  ->  ( ps  ->  E. x ph ) )
 
Theoremspcgf 2652 Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 2-Feb-1997.) (Revised by Andrew Salmon, 12-Aug-2011.)
 |-  F/_ x A   &    |-  F/ x ps   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A  e.  V  ->  ( A. x ph 
 ->  ps ) )
 
Theoremspcegf 2653 Existential specialization, using implicit substitution. (Contributed by NM, 2-Feb-1997.)
 |-  F/_ x A   &    |-  F/ x ps   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A  e.  V  ->  ( ps  ->  E. x ph ) )
 
Theoremspcimdv 2654* Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  A  e.  B )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  ( ps  ->  ch )
 )   =>    |-  ( ph  ->  ( A. x ps  ->  ch )
 )
 
Theoremspcdv 2655* Rule of specialization, using implicit substitution. Analogous to rspcdv 2676. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  B )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  ->  ch ) )
 
Theoremspcimedv 2656* Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  A  e.  B )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  ( ch  ->  ps )
 )   =>    |-  ( ph  ->  ( ch  ->  E. x ps )
 )
 
Theoremspcgv 2657* Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  V  ->  ( A. x ph 
 ->  ps ) )
 
Theoremspcegv 2658* Existential specialization, using implicit substitution. (Contributed by NM, 14-Aug-1994.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  V  ->  ( ps  ->  E. x ph ) )
 
Theoremspc2egv 2659* Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
 |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ps  ->  E. x E. y ph ) )
 
Theoremspc2gv 2660* Specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 27-Apr-2004.)
 |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. x A. y ph  ->  ps ) )
 
Theoremspc3egv 2661* Existential specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)
 |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph 
 <->  ps ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ps 
 ->  E. x E. y E. z ph ) )
 
Theoremspc3gv 2662* Specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)
 |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph 
 <->  ps ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A. x A. y A. z ph  ->  ps ) )
 
Theoremspcv 2663* Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ph 
 ->  ps )
 
Theoremspcev 2664* Existential specialization, using implicit substitution. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( ps  ->  E. x ph )
 
Theoremspc2ev 2665* Existential specialization, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   =>    |-  ( ps  ->  E. x E. y ph )
 
Theoremrspct 2666* A closed version of rspc 2667. (Contributed by Andrew Salmon, 6-Jun-2011.)
 |- 
 F/ x ps   =>    |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  ( A  e.  B  ->  ( A. x  e.  B  ph  ->  ps )
 ) )
 
Theoremrspc 2667* Restricted specialization, using implicit substitution. (Contributed by NM, 19-Apr-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)
 |- 
 F/ x ps   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A  e.  B  ->  ( A. x  e.  B  ph  ->  ps )
 )
 
Theoremrspce 2668* Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) (Revised by Mario Carneiro, 11-Oct-2016.)
 |- 
 F/ x ps   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( ( A  e.  B  /\  ps )  ->  E. x  e.  B  ph )
 
Theoremrspcv 2669* Restricted specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  B  ->  ( A. x  e.  B  ph  ->  ps )
 )
 
Theoremrspccv 2670* Restricted specialization, using implicit substitution. (Contributed by NM, 2-Feb-2006.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  e.  B  ph  ->  ( A  e.  B  ->  ps )
 )
 
Theoremrspcva 2671* Restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-2005.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( ( A  e.  B  /\  A. x  e.  B  ph )  ->  ps )
 
Theoremrspccva 2672* Restricted specialization, using implicit substitution. (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( ( A. x  e.  B  ph  /\  A  e.  B )  ->  ps )
 
Theoremrspcev 2673* Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( ( A  e.  B  /\  ps )  ->  E. x  e.  B  ph )
 
Theoremrspcimdv 2674* Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  A  e.  B )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  ( ps  ->  ch )
 )   =>    |-  ( ph  ->  ( A. x  e.  B  ps  ->  ch ) )
 
Theoremrspcimedv 2675* Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  A  e.  B )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  ( ch  ->  ps )
 )   =>    |-  ( ph  ->  ( ch  ->  E. x  e.  B  ps ) )
 
Theoremrspcdv 2676* Restricted specialization, using implicit substitution. (Contributed by NM, 17-Feb-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  A  e.  B )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  B  ps  ->  ch ) )
 
Theoremrspcedv 2677* Restricted existential specialization, using implicit substitution. (Contributed by FL, 17-Apr-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  A  e.  B )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( ch  ->  E. x  e.  B  ps ) )
 
Theoremrspcda 2678* Restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 29-Jun-2020.)
 |-  ( x  =  C  ->  ( ps  <->  ch ) )   &    |-  ( ph  ->  A. x  e.  A  ps )   &    |-  ( ph  ->  C  e.  A )   &    |-  F/ x ph   =>    |-  ( ph  ->  ch )
 
Theoremrspcdva 2679* Restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 21-Jun-2020.)
 |-  ( x  =  C  ->  ( ps  <->  ch ) )   &    |-  ( ph  ->  A. x  e.  A  ps )   &    |-  ( ph  ->  C  e.  A )   =>    |-  ( ph  ->  ch )
 
Theoremrspcedvd 2680* Restricted existential specialization, using implicit substitution. Variant of rspcedv 2677. (Contributed by AV, 27-Nov-2019.)
 |-  ( ph  ->  A  e.  B )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  ( ps  <->  ch ) )   &    |-  ( ph  ->  ch )   =>    |-  ( ph  ->  E. x  e.  B  ps )
 
Theoremrspcedeq1vd 2681* Restricted existential specialization, using implicit substitution. Variant of rspcedvd 2680 for equations, in which the left hand side depends on the quantified variable. (Contributed by AV, 24-Dec-2019.)
 |-  ( ph  ->  A  e.  B )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  C  =  D )   =>    |-  ( ph  ->  E. x  e.  B  C  =  D )
 
Theoremrspcedeq2vd 2682* Restricted existential specialization, using implicit substitution. Variant of rspcedvd 2680 for equations, in which the right hand side depends on the quantified variable. (Contributed by AV, 24-Dec-2019.)
 |-  ( ph  ->  A  e.  B )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  C  =  D )   =>    |-  ( ph  ->  E. x  e.  B  C  =  D )
 
Theoremrspc2 2683* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 9-Nov-2012.)
 |- 
 F/ x ch   &    |-  F/ y ps   &    |-  ( x  =  A  ->  ( ph  <->  ch ) )   &    |-  ( y  =  B  ->  ( ch  <->  ps ) )   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A. x  e.  C  A. y  e.  D  ph  ->  ps )
 )
 
Theoremrspc2gv 2684* Restricted specialization with two quantifiers, using implicit substitution. (Contributed by BJ, 2-Dec-2021.)
 |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. x  e.  V  A. y  e.  W  ph  ->  ps ) )
 
Theoremrspc2v 2685* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-1999.)
 |-  ( x  =  A  ->  ( ph  <->  ch ) )   &    |-  (
 y  =  B  ->  ( ch  <->  ps ) )   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A. x  e.  C  A. y  e.  D  ph  ->  ps ) )
 
Theoremrspc2va 2686* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 18-Jun-2014.)
 |-  ( x  =  A  ->  ( ph  <->  ch ) )   &    |-  (
 y  =  B  ->  ( ch  <->  ps ) )   =>    |-  ( ( ( A  e.  C  /\  B  e.  D )  /\  A. x  e.  C  A. y  e.  D  ph )  ->  ps )
 
Theoremrspc2ev 2687* 2-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 16-Oct-1999.)
 |-  ( x  =  A  ->  ( ph  <->  ch ) )   &    |-  (
 y  =  B  ->  ( ch  <->  ps ) )   =>    |-  ( ( A  e.  C  /\  B  e.  D  /\  ps )  ->  E. x  e.  C  E. y  e.  D  ph )
 
Theoremrspc3v 2688* 3-variable restricted specialization, using implicit substitution. (Contributed by NM, 10-May-2005.)
 |-  ( x  =  A  ->  ( ph  <->  ch ) )   &    |-  (
 y  =  B  ->  ( ch  <->  th ) )   &    |-  (
 z  =  C  ->  ( th  <->  ps ) )   =>    |-  ( ( A  e.  R  /\  B  e.  S  /\  C  e.  T )  ->  ( A. x  e.  R  A. y  e.  S  A. z  e.  T  ph  ->  ps )
 )
 
Theoremrspc3ev 2689* 3-variable restricted existentional specialization, using implicit substitution. (Contributed by NM, 25-Jul-2012.)
 |-  ( x  =  A  ->  ( ph  <->  ch ) )   &    |-  (
 y  =  B  ->  ( ch  <->  th ) )   &    |-  (
 z  =  C  ->  ( th  <->  ps ) )   =>    |-  ( ( ( A  e.  R  /\  B  e.  S  /\  C  e.  T )  /\  ps )  ->  E. x  e.  R  E. y  e.  S  E. z  e.  T  ph )
 
Theoremeqvinc 2690* A variable introduction law for class equality. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 |-  A  e.  _V   =>    |-  ( A  =  B 
 <-> 
 E. x ( x  =  A  /\  x  =  B ) )
 
Theoremeqvincg 2691* A variable introduction law for class equality, deduction version. (Contributed by Thierry Arnoux, 2-Mar-2017.)
 |-  ( A  e.  V  ->  ( A  =  B  <->  E. x ( x  =  A  /\  x  =  B ) ) )
 
Theoremeqvincf 2692 A variable introduction law for class equality, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Sep-2003.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  A  e.  _V   =>    |-  ( A  =  B  <->  E. x ( x  =  A  /\  x  =  B ) )
 
Theoremalexeq 2693* Two ways to express substitution of 
A for  x in  ph. (Contributed by NM, 2-Mar-1995.)
 |-  A  e.  _V   =>    |-  ( A. x ( x  =  A  -> 
 ph )  <->  E. x ( x  =  A  /\  ph )
 )
 
Theoremceqex 2694* Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.)
 |-  ( x  =  A  ->  ( ph  <->  E. x ( x  =  A  /\  ph )
 ) )
 
Theoremceqsexg 2695* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 11-Oct-2004.)
 |- 
 F/ x ps   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A  e.  V  ->  ( E. x ( x  =  A  /\  ph )  <->  ps ) )
 
Theoremceqsexgv 2696* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  V  ->  ( E. x ( x  =  A  /\  ph )  <->  ps ) )
 
Theoremceqsrexv 2697* Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 30-Apr-2004.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  B  ->  ( E. x  e.  B  ( x  =  A  /\  ph )  <->  ps ) )
 
Theoremceqsrexbv 2698* Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by Mario Carneiro, 14-Mar-2014.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( E. x  e.  B  ( x  =  A  /\  ph )  <->  ( A  e.  B  /\  ps ) )
 
Theoremceqsrex2v 2699* Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 29-Oct-2005.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( E. x  e.  C  E. y  e.  D  ( ( x  =  A  /\  y  =  B )  /\  ph )  <->  ch ) )
 
Theoremclel2 2700* An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
 |-  A  e.  _V   =>    |-  ( A  e.  B 
 <-> 
 A. x ( x  =  A  ->  x  e.  B ) )
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