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Theorem List for Metamath Proof Explorer - 2801-2900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelex22 2801* If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.)
 |-  ( ( A  e.  B  /\  A  e.  C )  ->  E. x ( x  e.  B  /\  x  e.  C ) )
 
Theoremelex2 2802* If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.)
 |-  ( A  e.  B  ->  E. x  x  e.  B )
 
Theoremralv 2803 A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
 |-  ( A. x  e. 
 _V  ph  <->  A. x ph )
 
Theoremrexv 2804 An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
 |-  ( E. x  e. 
 _V  ph  <->  E. x ph )
 
Theoremreuv 2805 A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.)
 |-  ( E! x  e. 
 _V  ph  <->  E! x ph )
 
Theoremrmov 2806 A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 |-  ( E* x  e. 
 _V ph  <->  E* x ph )
 
Theoremrabab 2807 A class abstraction restricted to the universe is unrestricted. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 |- 
 { x  e.  _V  |  ph }  =  { x  |  ph }
 
Theoremralcom4 2808* Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 |-  ( A. x  e.  A  A. y ph  <->  A. y A. x  e.  A  ph )
 
Theoremrexcom4 2809* Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 |-  ( E. x  e.  A  E. y ph  <->  E. y E. x  e.  A  ph )
 
Theoremrexcom4a 2810* Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
 |-  ( E. x E. y  e.  A  ( ph  /\  ps )  <->  E. y  e.  A  ( ph  /\  E. x ps ) )
 
Theoremrexcom4b 2811* Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
 |-  B  e.  _V   =>    |-  ( E. x E. y  e.  A  ( ph  /\  x  =  B )  <->  E. y  e.  A  ph )
 
Theoremceqsalt 2812* Closed theorem version of ceqsalg 2814. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
 |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  (
 ph 
 <->  ps ) )  /\  A  e.  V )  ->  ( A. x ( x  =  A  ->  ph )  <->  ps ) )
 
Theoremceqsralt 2813* Restricted quantifier version of ceqsalt 2812. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
 |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  (
 ph 
 <->  ps ) )  /\  A  e.  B )  ->  ( A. x  e.  B  ( x  =  A  ->  ph )  <->  ps ) )
 
Theoremceqsalg 2814* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 29-Oct-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 |- 
 F/ x ps   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A  e.  V  ->  ( A. x ( x  =  A  -> 
 ph )  <->  ps ) )
 
Theoremceqsal 2815* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)
 |- 
 F/ x ps   &    |-  A  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A. x ( x  =  A  ->  ph )  <->  ps )
 
Theoremceqsalv 2816* 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 2817* Restricted quantifier version of ceqsalv 2816. (Contributed by NM, 21-Jun-2013.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  B  ->  ( A. x  e.  B  ( x  =  A  ->  ph )  <->  ps ) )
 
Theoremgencl 2818* 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 2819* 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 2820* 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 2821* 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 2822* 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 2823* 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 2824* 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 2825* 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 2826* 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 2827* 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 2828* 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 2829* 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 2830* 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 2831* 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 2832* 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 2833* Restatement of gencbvex 2832 with weaker hypotheses. (Contributed by Jeffrey 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 2834* 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 2835* Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3118. (Contributed by Raph Levien, 10-Apr-2004.)
 |- 
 F/ x ps   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( y  =  A  ->  ( [
 y  /  x ] ph 
 <->  ps ) )
 
Theoremvtoclgft 2836 Closed theorem form of vtoclgf 2844. (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 2837 Implicit substitution of a class for a set 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 2838* Implicit substitution of a class for a set variable. (Contributed by Mario Carneiro, 15-Oct-2016.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  ( ps  <->  ch ) )   &    |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ch )
 
Theoremvtoclf 2839* Implicit substitution of a class for a set variable. This is a generalization of chvar 1929. (Contributed by NM, 30-Aug-1993.)
 |- 
 F/ x ps   &    |-  A  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ph   =>    |- 
 ps
 
Theoremvtocl 2840* Implicit substitution of a class for a set variable. (Contributed by NM, 30-Aug-1993.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  ph   =>    |- 
 ps
 
Theoremvtocl2 2841* Implicit substitution of classes for set 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 2842* Implicit substitution of classes for set 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 2843* Implicit substitution of a class for a set variable. (Contributed by NM, 23-Dec-1993.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  A  ->  ( ps  <->  th ) )   &    |-  ( ph 
 <->  ps )   =>    |-  ( ch  <->  th )
 
Theoremvtoclgf 2844 Implicit substitution of a class for a set 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 2845* Implicit substitution of a class expression for a set variable. (Contributed by NM, 17-Apr-1995.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ph   =>    |-  ( A  e.  V  ->  ps )
 
Theoremvtoclbg 2846* Implicit substitution of a class for a set variable. (Contributed by NM, 29-Apr-1994.)
 |-  ( x  =  A  ->  ( ph  <->  ch ) )   &    |-  ( x  =  A  ->  ( ps  <->  th ) )   &    |-  ( ph 
 <->  ps )   =>    |-  ( A  e.  V  ->  ( ch  <->  th ) )
 
Theoremvtocl2gf 2847 Implicit substitution of a class for a set 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 2848 Implicit substitution of a class for a set 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 2849* Implicit substitution of 2 classes for 2 set variables. (Contributed by NM, 25-Apr-1995.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   &    |-  ph   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ch )
 
Theoremvtoclgaf 2850* Implicit substitution of a class for a set 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 2851* Implicit substitution of a class for a set variable. (Contributed by NM, 20-Aug-1995.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  e.  B  ->  ph )   =>    |-  ( A  e.  B  ->  ps )
 
Theoremvtocl2gaf 2852* Implicit substitution of 2 classes for 2 set 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 2853* Implicit substitution of 2 classes for 2 set 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 2854* Implicit substitution of 3 classes for 3 set 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 2855* Implicit substitution of 3 classes for 3 set 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 2856* Implicit substitution of a class for a set variable. (Contributed by NM, 10-Jan-2004.)
 |-  ( x  =  A  -> 
 ph )   =>    |-  ( A  e.  V  -> 
 ph )
 
Theoremvtoclegft 2857* Implicit substitution of a class for a set variable. (Closed theorem version of vtoclef 2858.) (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 2858* Implicit substitution of a class for a set variable. (Contributed by NM, 18-Aug-1993.)
 |- 
 F/ x ph   &    |-  A  e.  _V   &    |-  ( x  =  A  ->  ph )   =>    |-  ph
 
Theoremvtocle 2859* Implicit substitution of a class for a set variable. (Contributed by NM, 9-Sep-1993.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  ph )   =>    |-  ph
 
Theoremvtoclri 2860* Implicit substitution of a class for a set variable. (Contributed by NM, 21-Nov-1994.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  A. x  e.  B  ph   =>    |-  ( A  e.  B  ->  ps )
 
Theoremspcimgft 2861 A closed version of spcimgf 2863. (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 2862 A closed version of spcgf 2865. (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 2863 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 2864 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 2865 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 2866 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 2867* 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 2868* Rule of specialization, using implicit substitution. Analogous to rspcdv 2889. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  B )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  ->  ch ) )
 
Theoremspcimedv 2869* 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 2870* 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 2871* Existential specialization, using implicit substitution. (Contributed by NM, 14-Aug-1994.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  V  ->  ( ps  ->  E. x ph ) )
 
Theoremspc2egv 2872* 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 2873* 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 2874* 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 2875* 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 2876* Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ph 
 ->  ps )
 
Theoremspcev 2877* 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 2878* 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 2879* A closed version of rspc 2880. (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 2880* 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 2881* 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 2882* 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 2883* 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 2884* 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 2885* 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 2886* 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 2887* 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 2888* 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 2889* 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 2890* 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 ) )
 
Theoremrspc2 2891* 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 2892* 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 2893* 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 2894* 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 2895* 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 2896* 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 2897* 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 2898 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 2899* 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 2900* Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.)
 |-  ( x  =  A  ->  ( ph  <->  E. x ( x  =  A  /\  ph )
 ) )
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