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Theorem List for Intuitionistic Logic Explorer - 2501-2600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremr19.28av 2501* Restricted version of one direction of Theorem 19.28 of [Margaris] p. 90. (The other direction doesn't hold when  A is empty.) (Contributed by NM, 2-Apr-2004.)
 |-  ( ( ph  /\  A. x  e.  A  ps )  ->  A. x  e.  A  ( ph  /\  ps )
 )
 
Theoremr19.29 2502 Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.) (Proof shortened by Andrew Salmon, 30-May-2011.)
 |-  ( ( A. x  e.  A  ph  /\  E. x  e.  A  ps )  ->  E. x  e.  A  ( ph  /\  ps )
 )
 
Theoremr19.29r 2503 Variation of Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.)
 |-  ( ( E. x  e.  A  ph  /\  A. x  e.  A  ps )  ->  E. x  e.  A  ( ph  /\  ps )
 )
 
Theoremr19.29af2 2504 A commonly used pattern based on r19.29 2502 (Contributed by Thierry Arnoux, 17-Dec-2017.)
 |- 
 F/ x ph   &    |-  F/ x ch   &    |-  (
 ( ( ph  /\  x  e.  A )  /\  ps )  ->  ch )   &    |-  ( ph  ->  E. x  e.  A  ps )   =>    |-  ( ph  ->  ch )
 
Theoremr19.29af 2505* A commonly used pattern based on r19.29 2502 (Contributed by Thierry Arnoux, 29-Nov-2017.)
 |- 
 F/ x ph   &    |-  ( ( (
 ph  /\  x  e.  A )  /\  ps )  ->  ch )   &    |-  ( ph  ->  E. x  e.  A  ps )   =>    |-  ( ph  ->  ch )
 
Theoremr19.29an 2506* A commonly used pattern based on r19.29 2502. (Contributed by Thierry Arnoux, 29-Dec-2019.)
 |-  ( ( ( ph  /\  x  e.  A ) 
 /\  ps )  ->  ch )   =>    |-  (
 ( ph  /\  E. x  e.  A  ps )  ->  ch )
 
Theoremr19.29a 2507* A commonly used pattern based on r19.29 2502 (Contributed by Thierry Arnoux, 22-Nov-2017.)
 |-  ( ( ( ph  /\  x  e.  A ) 
 /\  ps )  ->  ch )   &    |-  ( ph  ->  E. x  e.  A  ps )   =>    |-  ( ph  ->  ch )
 
Theoremr19.29d2r 2508 Theorem 19.29 of [Margaris] p. 90 with two restricted quantifiers, deduction version (Contributed by Thierry Arnoux, 30-Jan-2017.)
 |-  ( ph  ->  A. x  e.  A  A. y  e.  B  ps )   &    |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ch )   =>    |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ( ps  /\  ch ) )
 
Theoremr19.29vva 2509* A commonly used pattern based on r19.29 2502, version with two restricted quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.)
 |-  ( ( ( (
 ph  /\  x  e.  A )  /\  y  e.  B )  /\  ps )  ->  ch )   &    |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ps )   =>    |-  ( ph  ->  ch )
 
Theoremr19.32r 2510 One direction of Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers. For decidable propositions this is an equivalence. (Contributed by Jim Kingdon, 19-Aug-2018.)
 |- 
 F/ x ph   =>    |-  ( ( ph  \/  A. x  e.  A  ps )  ->  A. x  e.  A  ( ph  \/  ps )
 )
 
Theoremr19.32vr 2511* One direction of Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers. For decidable propositions this is an equivalence, as seen at r19.32vdc 2512. (Contributed by Jim Kingdon, 19-Aug-2018.)
 |-  ( ( ph  \/  A. x  e.  A  ps )  ->  A. x  e.  A  ( ph  \/  ps )
 )
 
Theoremr19.32vdc 2512* Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers, where  ph is decidable. (Contributed by Jim Kingdon, 4-Jun-2018.)
 |-  (DECID 
 ph  ->  ( A. x  e.  A  ( ph  \/  ps )  <->  ( ph  \/  A. x  e.  A  ps ) ) )
 
Theoremr19.35-1 2513 Restricted quantifier version of 19.35-1 1558. (Contributed by Jim Kingdon, 4-Jun-2018.)
 |-  ( E. x  e.  A  ( ph  ->  ps )  ->  ( A. x  e.  A  ph  ->  E. x  e.  A  ps ) )
 
Theoremr19.36av 2514* One direction of a restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. In classical logic, the converse would hold if  A has at least one element, but in intuitionistic logic, that is not a sufficient condition. (Contributed by NM, 22-Oct-2003.)
 |-  ( E. x  e.  A  ( ph  ->  ps )  ->  ( A. x  e.  A  ph  ->  ps ) )
 
Theoremr19.37 2515 Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. In classical logic the converse would hold if  A has at least one element, but that is not sufficient in intuitionistic logic. (Contributed by FL, 13-May-2012.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |- 
 F/ x ph   =>    |-  ( E. x  e.  A  ( ph  ->  ps )  ->  ( ph  ->  E. x  e.  A  ps ) )
 
Theoremr19.37av 2516* Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 2-Apr-2004.)
 |-  ( E. x  e.  A  ( ph  ->  ps )  ->  ( ph  ->  E. x  e.  A  ps ) )
 
Theoremr19.40 2517 Restricted quantifier version of Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 2-Apr-2004.)
 |-  ( E. x  e.  A  ( ph  /\  ps )  ->  ( E. x  e.  A  ph  /\  E. x  e.  A  ps ) )
 
Theoremr19.41 2518 Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 1-Nov-2010.)
 |- 
 F/ x ps   =>    |-  ( E. x  e.  A  ( ph  /\  ps ) 
 <->  ( E. x  e.  A  ph  /\  ps )
 )
 
Theoremr19.41v 2519* Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 17-Dec-2003.)
 |-  ( E. x  e.  A  ( ph  /\  ps ) 
 <->  ( E. x  e.  A  ph  /\  ps )
 )
 
Theoremr19.42v 2520* Restricted version of Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
 |-  ( E. x  e.  A  ( ph  /\  ps ) 
 <->  ( ph  /\  E. x  e.  A  ps ) )
 
Theoremr19.43 2521 Restricted version of Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) (Proof rewritten by Jim Kingdon, 5-Jun-2018.)
 |-  ( E. x  e.  A  ( ph  \/  ps )  <->  ( E. x  e.  A  ph  \/  E. x  e.  A  ps ) )
 
Theoremr19.44av 2522* One direction of a restricted quantifier version of Theorem 19.44 of [Margaris] p. 90. The other direction doesn't hold when  A is empty. (Contributed by NM, 2-Apr-2004.)
 |-  ( E. x  e.  A  ( ph  \/  ps )  ->  ( E. x  e.  A  ph  \/  ps ) )
 
Theoremr19.45av 2523* Restricted version of one direction of Theorem 19.45 of [Margaris] p. 90. (The other direction doesn't hold when  A is empty.) (Contributed by NM, 2-Apr-2004.)
 |-  ( E. x  e.  A  ( ph  \/  ps )  ->  ( ph  \/  E. x  e.  A  ps ) )
 
Theoremralcomf 2524* Commutation of restricted quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ y A   &    |-  F/_ x B   =>    |-  ( A. x  e.  A  A. y  e.  B  ph  <->  A. y  e.  B  A. x  e.  A  ph )
 
Theoremrexcomf 2525* Commutation of restricted quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ y A   &    |-  F/_ x B   =>    |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. y  e.  B  E. x  e.  A  ph )
 
Theoremralcom 2526* Commutation of restricted quantifiers. (Contributed by NM, 13-Oct-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
 |-  ( A. x  e.  A  A. y  e.  B  ph  <->  A. y  e.  B  A. x  e.  A  ph )
 
Theoremrexcom 2527* Commutation of restricted quantifiers. (Contributed by NM, 19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
 |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. y  e.  B  E. x  e.  A  ph )
 
Theoremrexcom13 2528* Swap 1st and 3rd restricted existential quantifiers. (Contributed by NM, 8-Apr-2015.)
 |-  ( E. x  e.  A  E. y  e.  B  E. z  e.  C  ph  <->  E. z  e.  C  E. y  e.  B  E. x  e.  A  ph )
 
Theoremrexrot4 2529* Rotate existential restricted quantifiers twice. (Contributed by NM, 8-Apr-2015.)
 |-  ( E. x  e.  A  E. y  e.  B  E. z  e.  C  E. w  e.  D  ph  <->  E. z  e.  C  E. w  e.  D  E. x  e.  A  E. y  e.  B  ph )
 
Theoremralcom3 2530 A commutative law for restricted quantifiers that swaps the domain of the restriction. (Contributed by NM, 22-Feb-2004.)
 |-  ( A. x  e.  A  ( x  e.  B  ->  ph )  <->  A. x  e.  B  ( x  e.  A  -> 
 ph ) )
 
Theoremreean 2531* Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Andrew Salmon, 30-May-2011.)
 |- 
 F/ y ph   &    |-  F/ x ps   =>    |-  ( E. x  e.  A  E. y  e.  B  ( ph  /\  ps )  <->  ( E. x  e.  A  ph 
 /\  E. y  e.  B  ps ) )
 
Theoremreeanv 2532* Rearrange existential quantifiers. (Contributed by NM, 9-May-1999.)
 |-  ( E. x  e.  A  E. y  e.  B  ( ph  /\  ps ) 
 <->  ( E. x  e.  A  ph  /\  E. y  e.  B  ps ) )
 
Theorem3reeanv 2533* Rearrange three existential quantifiers. (Contributed by Jeff Madsen, 11-Jun-2010.)
 |-  ( E. x  e.  A  E. y  e.  B  E. z  e.  C  ( ph  /\  ps  /\ 
 ch )  <->  ( E. x  e.  A  ph  /\  E. y  e.  B  ps  /\  E. z  e.  C  ch ) )
 
Theoremnfreu1 2534  x is not free in  E! x  e.  A ph. (Contributed by NM, 19-Mar-1997.)
 |- 
 F/ x E! x  e.  A  ph
 
Theoremnfrmo1 2535  x is not free in  E* x  e.  A ph. (Contributed by NM, 16-Jun-2017.)
 |- 
 F/ x E* x  e.  A  ph
 
Theoremnfreudxy 2536* Not-free deduction for restricted uniqueness. This is a version where  x and  y are distinct. (Contributed by Jim Kingdon, 6-Jun-2018.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x E! y  e.  A  ps )
 
Theoremnfreuxy 2537* 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 2538 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 2539* 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 2540 Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2603. (Contributed by NM, 25-Nov-2013.)
 |-  ( A. x  e.  A  ( ps  <->  ch )  <->  { x  e.  A  |  ps }  =  { x  e.  A  |  ch } )
 
Theoremrabswap 2541 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 2542 The abstraction variable in a restricted class abstraction isn't free. (Contributed by NM, 19-Mar-1997.)
 |-  F/_ x { x  e.  A  |  ph }
 
Theoremnfrabxy 2543* 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 2544 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 2545* 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 2546* 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 2547 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 2548 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 2549 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 2550* 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 2551* 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 2552 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 2553 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 2554 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 2555 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 2556 Equality theorem for restricted unique existential 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 2557 Equality theorem for restricted at-most-one 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 2558* 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 2559* 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 2560* Equality theorem for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.)
 |-  ( A  =  B  ->  ( E! x  e.  A  ph  <->  E! x  e.  B  ph ) )
 
Theoremrmoeq1 2561* Equality theorem for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 |-  ( A  =  B  ->  ( E* x  e.  A  ph  <->  E* x  e.  B  ph ) )
 
Theoremraleqi 2562* 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 2563* 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 2564* 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 2565* 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 2566* 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 2567* 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 2568* Equality deduction for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.)
 |-  ( A  =  B  ->  ( ph  <->  ps ) )   =>    |-  ( A  =  B  ->  ( E! x  e.  A  ph  <->  E! x  e.  B  ps ) )
 
Theoremrmoeqd 2569* Equality deduction for restricted at-most-one 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 2570* 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 2571* 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 2572* 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 2573* 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 2574 Unrestricted "at most one" implies restricted "at most one". (Contributed by NM, 16-Jun-2017.)
 |-  ( E* x ph  ->  E* x  e.  A  ph )
 
Theoremreu5 2575 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 2576 Restricted unique existence implies restricted existence. (Contributed by NM, 19-Aug-1999.)
 |-  ( E! x  e.  A  ph  ->  E. x  e.  A  ph )
 
Theoremreurmo 2577 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 2578 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 2579 Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.)
 |-  ( -.  E. x  e.  A  ph  ->  E* x  e.  A  ph )
 
Theoremcbvralf 2580 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 2581 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 2582* 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 2583* 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 2584* Change the bound variable of a restricted unique existential 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 2585* 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 2586* 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 2587* 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 2588* Change the bound variable of a restricted unique existential 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 2589* Change the bound variable of a restricted at-most-one 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 2590* 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 2591* 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 2592* 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 2593* 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 2594* 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 2595* 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 2596* 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 2597* 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 2598* 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 2599* 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 2600 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 }
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