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Theorem List for Metamath Proof Explorer - 3101-3200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremceqsex6v 3101* 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 3102* 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 3103* 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 3104* Restatement of gencbvex 3103 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 3105* Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.)
 |-  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 3106* Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3392. (Contributed by Raph Levien, 10-Apr-2004.)
 |- 
 F/ x ps   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( y  =  A  ->  ( [
 y  /  x ] ph 
 <->  ps ) )
 
Theoremvtoclgft 3107 Closed theorem form of vtoclgf 3115. (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 3108 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 3109* 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 3110* Implicit substitution of a class for a setvar variable. This is a generalization of chvar 2040. (Contributed by NM, 30-Aug-1993.)
 |- 
 F/ x ps   &    |-  A  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ph   =>    |- 
 ps
 
Theoremvtocl 3111* 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 3112* 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 3113* 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 3114* 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 3115 Implicit substitution of a class for a setvar variable, with bound-variable hypotheses in place of disjoint 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 )
 
Theoremvtoclg1f 3116* Version of vtoclgf 3115 with one non-freeness hypothesis replaced with a dv condition, thus avoiding dependency on ax-11 1866 and ax-13 2026. (Contributed by BJ, 1-May-2019.)
 |- 
 F/ x ps   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  ph   =>    |-  ( A  e.  V  ->  ps )
 
Theoremvtoclg 3117* 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 3118* 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 3119 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 3120 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 3121* 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 3122* 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 3123* 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 3124* 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 3125* 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 3126* 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 3127* 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 )
 
Theoremvtocl4g 3128* Implicit substitution of 4 classes for 4 setvar variables. (Contributed by AV, 22-Jan-2019.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 z  =  C  ->  ( ch  <->  rh ) )   &    |-  ( w  =  D  ->  ( rh  <->  th ) )   &    |-  ph   =>    |-  ( ( ( A  e.  Q  /\  B  e.  R )  /\  ( C  e.  S  /\  D  e.  T )
 )  ->  th )
 
Theoremvtocl4ga 3129* Implicit substitution of 4 classes for 4 setvar variables. (Contributed by AV, 22-Jan-2019.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 z  =  C  ->  ( ch  <->  rh ) )   &    |-  ( w  =  D  ->  ( rh  <->  th ) )   &    |-  (
 ( ( x  e.  Q  /\  y  e.  R )  /\  (
 z  e.  S  /\  w  e.  T )
 )  ->  ph )   =>    |-  ( ( ( A  e.  Q  /\  B  e.  R )  /\  ( C  e.  S  /\  D  e.  T ) )  ->  th )
 
Theoremvtocleg 3130* Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Jun-1993.)
 |-  ( x  =  A  -> 
 ph )   =>    |-  ( A  e.  V  -> 
 ph )
 
Theoremvtoclegft 3131* Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 3132.) (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 3132* 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 3133* Implicit substitution of a class for a setvar variable. (Contributed by NM, 9-Sep-1993.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  ph )   =>    |-  ph
 
Theoremvtoclri 3134* 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 3135 A closed version of spcimgf 3137. (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 3136 A closed version of spcgf 3139. (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 ) ) )
 
Theoremspcimgf 3137 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 3138 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 3139 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 3140 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 3141* 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 3142* Rule of specialization, using implicit substitution. Analogous to rspcdv 3163. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  B )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  ->  ch ) )
 
Theoremspcimedv 3143* 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 3144* 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 3145* Existential specialization, using implicit substitution. (Contributed by NM, 14-Aug-1994.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  V  ->  ( ps  ->  E. x ph ) )
 
Theoremspc2egv 3146* 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 3147* 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 3148* 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 3149* 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 3150* Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ph 
 ->  ps )
 
Theoremspcev 3151* 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 3152* 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 3153* A closed version of rspc 3154. (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 3154* 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 3155* 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 3156* 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 3157* 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 3158* 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 3159* 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 3160* 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 3161* 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 3162* 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 3163* 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 3164* 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 ) )
 
Theoremrspcedvd 3165* Restricted existential specialization, using implicit substitution. Variant of rspcdv 3163. (Contributed by AV, 27-Nov-2019.)
 |-  ( ph  ->  A  e.  B )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  ( ps  <->  ch ) )   &    |-  ( ph  ->  ch )   =>    |-  ( ph  ->  E. x  e.  B  ps )
 
Theoremrspcedeq1vd 3166* Restricted existential specialization, using implicit substitution. Variant of rspcedvd 3165 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 3167* Restricted existential specialization, using implicit substitution. Variant of rspcedvd 3165 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 3168* 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 )
 )
 
Theoremrspc2v 3169* 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 3170* 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 3171* 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 3172* 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 3173* 3-variable restricted existential 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 )
 
Theoremralxpxfr2d 3174* Transfer a universal quantifier between one variable with pair-like semantics and two. (Contributed by Stefan O'Rear, 27-Feb-2015.)
 |-  A  e.  _V   &    |-  ( ph  ->  ( x  e.  B  <->  E. y  e.  C  E. z  e.  D  x  =  A )
 )   &    |-  ( ( ph  /\  x  =  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  B  ps 
 <-> 
 A. y  e.  C  A. z  e.  D  ch ) )
 
Theoremrexraleqim 3175* Statement following from existence and generalization with equality. (Contributed by AV, 9-Feb-2019.)
 |-  ( x  =  z 
 ->  ( ps  <->  ph ) )   &    |-  (
 z  =  Y  ->  (
 ph 
 <-> 
 th ) )   =>    |-  ( ( E. z  e.  A  ph  /\  A. x  e.  A  ( ps  ->  x  =  Y ) )  ->  th )
 
Theoremeqvinc 3176* 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 ) )
 
Theoremeqvincf 3177 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 ) )
 
Theoremalexeqg 3178* Two ways to express substitution of 
A for  x in  ph. This is the analogue for classes of sb56 2196. (Contributed by NM, 2-Mar-1995.) (Revised by BJ, 27-Apr-2019.)
 |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  ph )  <->  E. x ( x  =  A  /\  ph )
 ) )
 
Theoremalexeq 3179* Two ways to express substitution of 
A for  x in  ph. Obsoleted by alexeqg 3178. (Contributed by NM, 2-Mar-1995.) Obsolete as of 1-May-2019. (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( A. x ( x  =  A  -> 
 ph )  <->  E. x ( x  =  A  /\  ph )
 )
 
Theoremceqex 3180* Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.) (Proof shortened by BJ, 1-May-2019.)
 |-  ( x  =  A  ->  ( ph  <->  E. x ( x  =  A  /\  ph )
 ) )
 
Theoremceqsexg 3181* 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 3182* 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 3183* 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 3184* 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 3185* 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 3186* 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 ) )
 
Theoremclel3g 3187* An alternate definition of class membership when the class is a set. (Contributed by NM, 13-Aug-2005.)
 |-  ( B  e.  V  ->  ( A  e.  B  <->  E. x ( x  =  B  /\  A  e.  x ) ) )
 
Theoremclel3 3188* An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
 |-  B  e.  _V   =>    |-  ( A  e.  B 
 <-> 
 E. x ( x  =  B  /\  A  e.  x ) )
 
Theoremclel4 3189* An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
 |-  B  e.  _V   =>    |-  ( A  e.  B 
 <-> 
 A. x ( x  =  B  ->  A  e.  x ) )
 
Theorempm13.183 3190* Compare theorem *13.183 in [WhiteheadRussell] p. 178. Only  A is required to be a set. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  ( A  e.  V  ->  ( A  =  B  <->  A. z ( z  =  A  <->  z  =  B ) ) )
 
Theoremrr19.3v 3191* Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. We don't need the nonempty class condition of r19.3rzv 3866 when there is an outer quantifier. (Contributed by NM, 25-Oct-2012.)
 |-  ( A. x  e.  A  A. y  e.  A  ph  <->  A. x  e.  A  ph )
 
Theoremrr19.28v 3192* Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. We don't need the nonempty class condition of r19.28zv 3868 when there is an outer quantifier. (Contributed by NM, 29-Oct-2012.)
 |-  ( A. x  e.  A  A. y  e.  A  ( ph  /\  ps ) 
 <-> 
 A. x  e.  A  ( ph  /\  A. y  e.  A  ps ) )
 
Theoremelabgt 3193* Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 3197.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 |-  ( ( A  e.  B  /\  A. x ( x  =  A  ->  (
 ph 
 <->  ps ) ) ) 
 ->  ( A  e.  { x  |  ph }  <->  ps ) )
 
Theoremelabgf 3194 Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Revised by Mario Carneiro, 12-Oct-2016.)
 |-  F/_ x A   &    |-  F/ x ps   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A  e.  B  ->  ( A  e.  { x  |  ph }  <->  ps ) )
 
Theoremelabf 3195* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 12-Oct-2016.)
 |- 
 F/ x ps   &    |-  A  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  { x  |  ph }  <->  ps )
 
Theoremelab 3196* Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A  e.  { x  |  ph }  <->  ps )
 
Theoremelabg 3197* Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  V  ->  ( A  e.  { x  |  ph }  <->  ps ) )
 
Theoremelab2g 3198* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  B  =  { x  |  ph }   =>    |-  ( A  e.  V  ->  ( A  e.  B  <->  ps ) )
 
Theoremelab2 3199* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  B  =  { x  |  ph }   =>    |-  ( A  e.  B  <->  ps )
 
Theoremelab4g 3200* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  B  =  { x  |  ph }   =>    |-  ( A  e.  B  <->  ( A  e.  _V  /\  ps ) )
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