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Theorem List for Intuitionistic Logic Explorer - 2501-2600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnfreuxy 2501* Not-free for restricted uniqueness. This is a version where  x and  y are distinct. (Contributed by Jim Kingdon, 6-Jun-2018.)
 |-  F/_ x A   &    |-  F/ x ph   =>    |-  F/ x E! y  e.  A  ph
 
Theoremrabid 2502 An "identity" law of concretion for restricted abstraction. Special case of Definition 2.1 of [Quine] p. 16. (Contributed by NM, 9-Oct-2003.)
 |-  ( x  e.  { x  e.  A  |  ph
 } 
 <->  ( x  e.  A  /\  ph ) )
 
Theoremrabid2 2503* An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
 |-  ( A  =  { x  e.  A  |  ph
 } 
 <-> 
 A. x  e.  A  ph )
 
Theoremrabbi 2504 Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2565. (Contributed by NM, 25-Nov-2013.)
 |-  ( A. x  e.  A  ( ps  <->  ch )  <->  { x  e.  A  |  ps }  =  { x  e.  A  |  ch } )
 
Theoremrabswap 2505 Swap with a membership relation in a restricted class abstraction. (Contributed by NM, 4-Jul-2005.)
 |- 
 { x  e.  A  |  x  e.  B }  =  { x  e.  B  |  x  e.  A }
 
Theoremnfrab1 2506 The abstraction variable in a restricted class abstraction isn't free. (Contributed by NM, 19-Mar-1997.)
 |-  F/_ x { x  e.  A  |  ph }
 
Theoremnfrabxy 2507* A variable not free in a wff remains so in a restricted class abstraction. (Contributed by Jim Kingdon, 19-Jul-2018.)
 |- 
 F/ x ph   &    |-  F/_ x A   =>    |-  F/_ x { y  e.  A  |  ph }
 
Theoremreubida 2508 Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by Mario Carneiro, 19-Nov-2016.)
 |- 
 F/ x ph   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E! x  e.  A  ps 
 <->  E! x  e.  A  ch ) )
 
Theoremreubidva 2509* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 13-Nov-2004.)
 |-  ( ( ph  /\  x  e.  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E! x  e.  A  ps 
 <->  E! x  e.  A  ch ) )
 
Theoremreubidv 2510* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 17-Oct-1996.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E! x  e.  A  ps 
 <->  E! x  e.  A  ch ) )
 
Theoremreubiia 2511 Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 14-Nov-2004.)
 |-  ( x  e.  A  ->  ( ph  <->  ps ) )   =>    |-  ( E! x  e.  A  ph  <->  E! x  e.  A  ps )
 
Theoremreubii 2512 Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 22-Oct-1999.)
 |-  ( ph  <->  ps )   =>    |-  ( E! x  e.  A  ph  <->  E! x  e.  A  ps )
 
Theoremrmobida 2513 Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
 |- 
 F/ x ph   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E* x  e.  A  ps 
 <->  E* x  e.  A  ch ) )
 
Theoremrmobidva 2514* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
 |-  ( ( ph  /\  x  e.  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E* x  e.  A  ps 
 <->  E* x  e.  A  ch ) )
 
Theoremrmobidv 2515* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E* x  e.  A  ps 
 <->  E* x  e.  A  ch ) )
 
Theoremrmobiia 2516 Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 16-Jun-2017.)
 |-  ( x  e.  A  ->  ( ph  <->  ps ) )   =>    |-  ( E* x  e.  A  ph  <->  E* x  e.  A  ps )
 
Theoremrmobii 2517 Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 16-Jun-2017.)
 |-  ( ph  <->  ps )   =>    |-  ( E* x  e.  A  ph  <->  E* x  e.  A  ps )
 
Theoremraleqf 2518 Equality theorem for restricted universal quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  ( A  =  B  ->  ( A. x  e.  A  ph  <->  A. x  e.  B  ph ) )
 
Theoremrexeqf 2519 Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon, 11-Jul-2011.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  ( A  =  B  ->  ( E. x  e.  A  ph  <->  E. x  e.  B  ph ) )
 
Theoremreueq1f 2520 Equality theorem for restricted uniqueness quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 5-Apr-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  ( A  =  B  ->  ( E! x  e.  A  ph  <->  E! x  e.  B  ph ) )
 
Theoremrmoeq1f 2521 Equality theorem for restricted uniqueness quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  ( A  =  B  ->  ( E* x  e.  A  ph  <->  E* x  e.  B  ph ) )
 
Theoremraleq 2522* Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
 |-  ( A  =  B  ->  ( A. x  e.  A  ph  <->  A. x  e.  B  ph ) )
 
Theoremrexeq 2523* Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.)
 |-  ( A  =  B  ->  ( E. x  e.  A  ph  <->  E. x  e.  B  ph ) )
 
Theoremreueq1 2524* Equality theorem for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)
 |-  ( A  =  B  ->  ( E! x  e.  A  ph  <->  E! x  e.  B  ph ) )
 
Theoremrmoeq1 2525* Equality theorem for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 |-  ( A  =  B  ->  ( E* x  e.  A  ph  <->  E* x  e.  B  ph ) )
 
Theoremraleqi 2526* Equality inference for restricted universal qualifier. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  A  =  B   =>    |-  ( A. x  e.  A  ph  <->  A. x  e.  B  ph )
 
Theoremrexeqi 2527* Equality inference for restricted existential qualifier. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  A  =  B   =>    |-  ( E. x  e.  A  ph  <->  E. x  e.  B  ph )
 
Theoremraleqdv 2528* Equality deduction for restricted universal quantifier. (Contributed by NM, 13-Nov-2005.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A. x  e.  A  ps 
 <-> 
 A. x  e.  B  ps ) )
 
Theoremrexeqdv 2529* Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( E. x  e.  A  ps 
 <-> 
 E. x  e.  B  ps ) )
 
Theoremraleqbi1dv 2530* Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
 |-  ( A  =  B  ->  ( ph  <->  ps ) )   =>    |-  ( A  =  B  ->  ( A. x  e.  A  ph  <->  A. x  e.  B  ps ) )
 
Theoremrexeqbi1dv 2531* Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.)
 |-  ( A  =  B  ->  ( ph  <->  ps ) )   =>    |-  ( A  =  B  ->  ( E. x  e.  A  ph  <->  E. x  e.  B  ps ) )
 
Theoremreueqd 2532* Equality deduction for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)
 |-  ( A  =  B  ->  ( ph  <->  ps ) )   =>    |-  ( A  =  B  ->  ( E! x  e.  A  ph  <->  E! x  e.  B  ps ) )
 
Theoremrmoeqd 2533* Equality deduction for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 |-  ( A  =  B  ->  ( ph  <->  ps ) )   =>    |-  ( A  =  B  ->  ( E* x  e.  A  ph  <->  E* x  e.  B  ps ) )
 
Theoremraleqbidv 2534* Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  A  ps 
 <-> 
 A. x  e.  B  ch ) )
 
Theoremrexeqbidv 2535* Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  A  ps 
 <-> 
 E. x  e.  B  ch ) )
 
Theoremraleqbidva 2536* Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  A  ps 
 <-> 
 A. x  e.  B  ch ) )
 
Theoremrexeqbidva 2537* Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  A  ps 
 <-> 
 E. x  e.  B  ch ) )
 
Theoremmormo 2538 Unrestricted "at most one" implies restricted "at most one". (Contributed by NM, 16-Jun-2017.)
 |-  ( E* x ph  ->  E* x  e.  A  ph )
 
Theoremreu5 2539 Restricted uniqueness in terms of "at most one." (Contributed by NM, 23-May-1999.) (Revised by NM, 16-Jun-2017.)
 |-  ( E! x  e.  A  ph  <->  ( E. x  e.  A  ph  /\  E* x  e.  A  ph ) )
 
Theoremreurex 2540 Restricted unique existence implies restricted existence. (Contributed by NM, 19-Aug-1999.)
 |-  ( E! x  e.  A  ph  ->  E. x  e.  A  ph )
 
Theoremreurmo 2541 Restricted existential uniqueness implies restricted "at most one." (Contributed by NM, 16-Jun-2017.)
 |-  ( E! x  e.  A  ph  ->  E* x  e.  A  ph )
 
Theoremrmo5 2542 Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.)
 |-  ( E* x  e.  A  ph  <->  ( E. x  e.  A  ph  ->  E! x  e.  A  ph ) )
 
Theoremnrexrmo 2543 Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.)
 |-  ( -.  E. x  e.  A  ph  ->  E* x  e.  A  ph )
 
Theoremcbvralf 2544 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro, 9-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ y A   &    |-  F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x  e.  A  ph  <->  A. y  e.  A  ps )
 
Theoremcbvrexf 2545 Rule used to change bound variables, using implicit substitution. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 9-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.)
 |-  F/_ x A   &    |-  F/_ y A   &    |-  F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x  e.  A  ph  <->  E. y  e.  A  ps )
 
Theoremcbvral 2546* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.)
 |- 
 F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x  e.  A  ph  <->  A. y  e.  A  ps )
 
Theoremcbvrex 2547* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 |- 
 F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x  e.  A  ph  <->  E. y  e.  A  ps )
 
Theoremcbvreu 2548* Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Mario Carneiro, 15-Oct-2016.)
 |- 
 F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E! x  e.  A  ph  <->  E! y  e.  A  ps )
 
Theoremcbvrmo 2549* Change the bound variable of restricted "at most one" using implicit substitution. (Contributed by NM, 16-Jun-2017.)
 |- 
 F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E* x  e.  A  ph  <->  E* y  e.  A  ps )
 
Theoremcbvralv 2550* Change the bound variable of a restricted universal quantifier using implicit substitution. (Contributed by NM, 28-Jan-1997.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x  e.  A  ph  <->  A. y  e.  A  ps )
 
Theoremcbvrexv 2551* Change the bound variable of a restricted existential quantifier using implicit substitution. (Contributed by NM, 2-Jun-1998.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E. x  e.  A  ph  <->  E. y  e.  A  ps )
 
Theoremcbvreuv 2552* Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E! x  e.  A  ph  <->  E! y  e.  A  ps )
 
Theoremcbvrmov 2553* Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E* x  e.  A  ph  <->  E* y  e.  A  ps )
 
Theoremcbvraldva2 2554* Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ( ph  /\  x  =  y )  ->  ( ps 
 <->  ch ) )   &    |-  (
 ( ph  /\  x  =  y )  ->  A  =  B )   =>    |-  ( ph  ->  ( A. x  e.  A  ps 
 <-> 
 A. y  e.  B  ch ) )
 
Theoremcbvrexdva2 2555* Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ( ph  /\  x  =  y )  ->  ( ps 
 <->  ch ) )   &    |-  (
 ( ph  /\  x  =  y )  ->  A  =  B )   =>    |-  ( ph  ->  ( E. x  e.  A  ps 
 <-> 
 E. y  e.  B  ch ) )
 
Theoremcbvraldva 2556* Rule used to change the bound variable in a restricted universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ( ph  /\  x  =  y )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  A  ps 
 <-> 
 A. y  e.  A  ch ) )
 
Theoremcbvrexdva 2557* Rule used to change the bound variable in a restricted existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ( ph  /\  x  =  y )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  A  ps 
 <-> 
 E. y  e.  A  ch ) )
 
Theoremcbvral2v 2558* Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by NM, 10-Aug-2004.)
 |-  ( x  =  z 
 ->  ( ph  <->  ch ) )   &    |-  (
 y  =  w  ->  ( ch  <->  ps ) )   =>    |-  ( A. x  e.  A  A. y  e.  B  ph  <->  A. z  e.  A  A. w  e.  B  ps )
 
Theoremcbvrex2v 2559* Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by FL, 2-Jul-2012.)
 |-  ( x  =  z 
 ->  ( ph  <->  ch ) )   &    |-  (
 y  =  w  ->  ( ch  <->  ps ) )   =>    |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. z  e.  A  E. w  e.  B  ps )
 
Theoremcbvral3v 2560* Change bound variables of triple restricted universal quantification, using implicit substitution. (Contributed by NM, 10-May-2005.)
 |-  ( x  =  w 
 ->  ( ph  <->  ch ) )   &    |-  (
 y  =  v  ->  ( ch  <->  th ) )   &    |-  (
 z  =  u  ->  ( th  <->  ps ) )   =>    |-  ( A. x  e.  A  A. y  e.  B  A. z  e.  C  ph  <->  A. w  e.  A  A. v  e.  B  A. u  e.  C  ps )
 
Theoremcbvralsv 2561* Change bound variable by using a substitution. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.)
 |-  ( A. x  e.  A  ph  <->  A. y  e.  A  [ y  /  x ] ph )
 
Theoremcbvrexsv 2562* Change bound variable by using a substitution. (Contributed by NM, 2-Mar-2008.) (Revised by Andrew Salmon, 11-Jul-2011.)
 |-  ( E. x  e.  A  ph  <->  E. y  e.  A  [ y  /  x ] ph )
 
Theoremsbralie 2563* Implicit to explicit substitution that swaps variables in a quantified expression. (Contributed by NM, 5-Sep-2004.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( [ x  /  y ] A. x  e.  y  ph  <->  A. y  e.  x  ps )
 
Theoremrabbiia 2564 Equivalent wff's yield equal restricted class abstractions (inference rule). (Contributed by NM, 22-May-1999.)
 |-  ( x  e.  A  ->  ( ph  <->  ps ) )   =>    |-  { x  e.  A  |  ph }  =  { x  e.  A  |  ps }
 
Theoremrabbidva 2565* Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 28-Nov-2003.)
 |-  ( ( ph  /\  x  e.  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  =  { x  e.  A  |  ch }
 )
 
Theoremrabbidv 2566* Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 10-Feb-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  =  { x  e.  A  |  ch }
 )
 
Theoremrabeqf 2567 Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  ( A  =  B  ->  { x  e.  A  |  ph }  =  { x  e.  B  |  ph } )
 
Theoremrabeq 2568* Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.)
 |-  ( A  =  B  ->  { x  e.  A  |  ph }  =  { x  e.  B  |  ph
 } )
 
Theoremrabeqbidv 2569* Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  =  { x  e.  B  |  ch }
 )
 
Theoremrabeqbidva 2570* Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  =  { x  e.  B  |  ch }
 )
 
Theoremrabeq2i 2571 Inference rule from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.)
 |-  A  =  { x  e.  B  |  ph }   =>    |-  ( x  e.  A  <->  ( x  e.  B  /\  ph )
 )
 
Theoremcbvrab 2572 Rule to change the bound variable in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 9-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ y A   &    |-  F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  { x  e.  A  |  ph }  =  { y  e.  A  |  ps }
 
Theoremcbvrabv 2573* Rule to change the bound variable in a restricted class abstraction, using implicit substitution. (Contributed by NM, 26-May-1999.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  { x  e.  A  |  ph }  =  { y  e.  A  |  ps }
 
2.1.6  The universal class
 
Syntaxcvv 2574 Extend class notation to include the universal class symbol.
 class  _V
 
Theoremvjust 2575 Soundness justification theorem for df-v 2576. (Contributed by Rodolfo Medina, 27-Apr-2010.)
 |- 
 { x  |  x  =  x }  =  {
 y  |  y  =  y }
 
Definitiondf-v 2576 Define the universal class. Definition 5.20 of [TakeutiZaring] p. 21. Also Definition 2.9 of [Quine] p. 19. (Contributed by NM, 5-Aug-1993.)
 |- 
 _V  =  { x  |  x  =  x }
 
Theoremvex 2577 All setvar variables are sets (see isset 2578). Theorem 6.8 of [Quine] p. 43. (Contributed by NM, 5-Aug-1993.)
 |-  x  e.  _V
 
Theoremisset 2578* Two ways to say " A is a set": A class  A is a member of the universal class  _V (see df-v 2576) if and only if the class  A exists (i.e. there exists some set  x equal to class 
A). Theorem 6.9 of [Quine] p. 43. Notational convention: We will use the notational device " A  e.  _V " to mean " A is a set" very frequently, for example in uniex 4202. Note the when  A is not a set, it is called a proper class. In some theorems, such as uniexg 4203, in order to shorten certain proofs we use the more general antecedent  A  e.  V instead of  A  e.  _V to mean " A is a set."

Note that a constant is implicitly considered distinct from all variables. This is why  _V is not included in the distinct variable list, even though df-clel 2052 requires that the expression substituted for  B not contain  x. (Also, the Metamath spec does not allow constants in the distinct variable list.) (Contributed by NM, 26-May-1993.)

 |-  ( A  e.  _V  <->  E. x  x  =  A )
 
Theoremissetf 2579 A version of isset that does not require x and A to be distinct. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 10-Oct-2016.)
 |-  F/_ x A   =>    |-  ( A  e.  _V  <->  E. x  x  =  A )
 
Theoremisseti 2580* A way to say " A is a set" (inference rule). (Contributed by NM, 5-Aug-1993.)
 |-  A  e.  _V   =>    |-  E. x  x  =  A
 
Theoremissetri 2581* A way to say " A is a set" (inference rule). (Contributed by NM, 5-Aug-1993.)
 |- 
 E. x  x  =  A   =>    |-  A  e.  _V
 
Theoremeqvisset 2582 A class equal to a variable is a set. Note the absence of dv condition, contrary to isset 2578 and issetri 2581. (Contributed by BJ, 27-Apr-2019.)
 |-  ( x  =  A  ->  A  e.  _V )
 
Theoremelex 2583 If a class is a member of another class, it is a set. Theorem 6.12 of [Quine] p. 44. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 |-  ( A  e.  B  ->  A  e.  _V )
 
Theoremelexi 2584 If a class is a member of another class, it is a set. (Contributed by NM, 11-Jun-1994.)
 |-  A  e.  B   =>    |-  A  e.  _V
 
Theoremelisset 2585* An element of a class exists. (Contributed by NM, 1-May-1995.)
 |-  ( A  e.  V  ->  E. x  x  =  A )
 
Theoremelex22 2586* 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 2587* 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 2588 A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
 |-  ( A. x  e. 
 _V  ph  <->  A. x ph )
 
Theoremrexv 2589 An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
 |-  ( E. x  e. 
 _V  ph  <->  E. x ph )
 
Theoremreuv 2590 A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.)
 |-  ( E! x  e. 
 _V  ph  <->  E! x ph )
 
Theoremrmov 2591 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 2592 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 2593* 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 2594* 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 2595* 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 2596* 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 2597* Closed theorem version of ceqsalg 2599. (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 2598* Restricted quantifier version of ceqsalt 2597. (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 2599* 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 2600* 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 )
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