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Theorem List for Metamath Proof Explorer - 2801-2900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsbhypf 2801* Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3077. (Contributed by Raph Levien, 10-Apr-2004.)
 |- 
 F/ x ps   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( y  =  A  ->  ( [
 y  /  x ] ph 
 <->  ps ) )
 
Theoremvtoclgft 2802 Closed theorem form of vtoclgf 2810. (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 2803 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 2804* 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 2805* Implicit substitution of a class for a set variable. This is a generalization of chvar 1879. (Contributed by NM, 30-Aug-1993.)
 |- 
 F/ x ps   &    |-  A  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ph   =>    |- 
 ps
 
Theoremvtocl 2806* 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 2807* 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 2808* 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 2809* 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 2810 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 2811* 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 2812* 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 2813 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 2814 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 2815* 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 2816* 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 2817* 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 2818* 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 2819* 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 2820* 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 2821* 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 2822* Implicit substitution of a class for a set variable. (Contributed by NM, 10-Jan-2004.)
 |-  ( x  =  A  -> 
 ph )   =>    |-  ( A  e.  V  -> 
 ph )
 
Theoremvtoclegft 2823* Implicit substitution of a class for a set variable. (Closed theorem version of vtoclef 2824.) (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 2824* 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 2825* Implicit substitution of a class for a set variable. (Contributed by NM, 9-Sep-1993.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  ph )   =>    |-  ph
 
Theoremvtoclri 2826* 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 )
 
Theoremcla4imgft 2827 A closed version of cla4imgf 2829. (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 ) ) )
 
Theoremcla4gft 2828 A closed version of cla4gf 2831. (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 ) ) )
 
Theoremcla4imgf 2829 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 ) )
 
Theoremcla4imegf 2830 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 ) )
 
Theoremcla4gf 2831 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 ) )
 
Theoremcla4egf 2832 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 ) )
 
Theoremcla4imdv 2833* 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 )
 )
 
Theoremcla4dv 2834* Rule of specialization, using implicit substitution. Analogous to rcla4dv 2855. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  B )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  ->  ch ) )
 
Theoremcla4imedv 2835* 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 )
 )
 
Theoremcla4gv 2836* 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 ) )
 
Theoremcla4egv 2837* Existential specialization, using implicit substitution. (Contributed by NM, 14-Aug-1994.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  V  ->  ( ps  ->  E. x ph ) )
 
Theoremcla42egv 2838* 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 ) )
 
Theoremcla42gv 2839* 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 ) )
 
Theoremcla43egv 2840* 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 ) )
 
Theoremcla43gv 2841* 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 ) )
 
Theoremcla4v 2842* Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ph 
 ->  ps )
 
Theoremcla4ev 2843* 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 )
 
Theoremcla42ev 2844* 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 )
 
Theoremrcla4t 2845* A closed version of rcla4 2846. (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 )
 ) )
 
Theoremrcla4 2846* 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 )
 )
 
Theoremrcla4e 2847* 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 )
 
Theoremrcla4v 2848* Restricted specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  B  ->  ( A. x  e.  B  ph  ->  ps )
 )
 
Theoremrcla4cv 2849* Restricted specialization, using implicit substitution. (Contributed by NM, 2-Feb-2006.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  e.  B  ph  ->  ( A  e.  B  ->  ps )
 )
 
Theoremrcla4va 2850* Restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-2005.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( ( A  e.  B  /\  A. x  e.  B  ph )  ->  ps )
 
Theoremrcla4cva 2851* 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 )
 
Theoremrcla4ev 2852* 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 )
 
Theoremrcla4imdv 2853* 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 ) )
 
Theoremrcla4imedv 2854* 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 ) )
 
Theoremrcla4dv 2855* 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 ) )
 
Theoremrcla4edv 2856* 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 ) )
 
Theoremrcla42 2857* 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 )
 )
 
Theoremrcla42v 2858* 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 ) )
 
Theoremrcla42va 2859* 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 )
 
Theoremrcla42ev 2860* 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 )
 
Theoremrcla43v 2861* 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 )
 )
 
Theoremrcla43ev 2862* 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 2863* 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 2864 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 2865* 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 2866* Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.)
 |-  ( x  =  A  ->  ( ph  <->  E. x ( x  =  A  /\  ph )
 ) )
 
Theoremceqsexg 2867* 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 2868* 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 2869* 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 2870* 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 2871* 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 2872* 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 2873* 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 2874* 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 2875* 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 2876* 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 2877* Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. We don't need the non-empty class condition of r19.3rzv 3508 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 2878* Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. We don't need the non-empty class condition of r19.28zv 3510 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 2879* Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 2883.) (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 2880 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 2881* 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 2882* 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 2883* 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 2884* 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 2885* 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 2886* 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 ) )
 
Theoremelab3gf 2887 Membership in a class abstraction, with a weaker antecedent than elabgf 2880. (Contributed by NM, 6-Sep-2011.)
 |-  F/_ x A   &    |-  F/ x ps   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( ( ps 
 ->  A  e.  B ) 
 ->  ( A  e.  { x  |  ph }  <->  ps ) )
 
Theoremelab3g 2888* Membership in a class abstraction, with a weaker antecedent than elabg 2883. (Contributed by NM, 29-Aug-2006.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( ( ps 
 ->  A  e.  B ) 
 ->  ( A  e.  { x  |  ph }  <->  ps ) )
 
Theoremelab3 2889* Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.)
 |-  ( ps  ->  A  e.  _V )   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  { x  |  ph }  <->  ps )
 
Theoremelrabf 2890 Membership in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  F/ x ps   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  { x  e.  B  |  ph
 } 
 <->  ( A  e.  B  /\  ps ) )
 
Theoremelrab 2891* Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 21-May-1999.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  { x  e.  B  |  ph
 } 
 <->  ( A  e.  B  /\  ps ) )
 
Theoremelrab3 2892* Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  B  ->  ( A  e.  { x  e.  B  |  ph
 } 
 <->  ps ) )
 
Theoremelrab2 2893* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  C  =  { x  e.  B  |  ph }   =>    |-  ( A  e.  C  <->  ( A  e.  B  /\  ps ) )
 
Theoremralab 2894* Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  ( y  =  x 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x  e.  { y  |  ph } ch  <->  A. x ( ps 
 ->  ch ) )
 
Theoremralrab 2895* Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  ( y  =  x 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x  e.  { y  e.  A  |  ph } ch  <->  A. x  e.  A  ( ps  ->  ch )
 )
 
Theoremrexab 2896* Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.)
 |-  ( y  =  x 
 ->  ( ph  <->  ps ) )   =>    |-  ( E. x  e.  { y  |  ph } ch  <->  E. x ( ps 
 /\  ch ) )
 
Theoremrexrab 2897* Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.)
 |-  ( y  =  x 
 ->  ( ph  <->  ps ) )   =>    |-  ( E. x  e.  { y  e.  A  |  ph } ch  <->  E. x  e.  A  ( ps  /\  ch )
 )
 
Theoremralab2 2898* Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( x  =  y 
 ->  ( ps  <->  ch ) )   =>    |-  ( A. x  e.  { y  |  ph } ps  <->  A. y ( ph  ->  ch ) )
 
Theoremralrab2 2899* Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( x  =  y 
 ->  ( ps  <->  ch ) )   =>    |-  ( A. x  e.  { y  e.  A  |  ph } ps  <->  A. y  e.  A  ( ph  ->  ch )
 )
 
Theoremrexab2 2900* Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( x  =  y 
 ->  ( ps  <->  ch ) )   =>    |-  ( E. x  e.  { y  |  ph } ps  <->  E. y ( ph  /\ 
 ch ) )
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