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Theorem List for Intuitionistic Logic Explorer - 2601-2700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremvtoclf 2601* Implicit substitution of a class for a setvar variable. This is a generalization of chvar 1637. (Contributed by NM, 30-Aug-1993.)

 F/   &     _V   &       &       =>   
 
Theoremvtocl 2602* Implicit substitution of a class for a setvar variable. (Contributed by NM, 30-Aug-1993.)
 _V   &       &       =>   
 
Theoremvtocl2 2603* Implicit substitution of classes for setvar variables. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 _V   &     _V   &       &       =>   
 
Theoremvtocl3 2604* Implicit substitution of classes for setvar variables. (Contributed by NM, 3-Jun-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 _V   &     _V   &     C  _V   &     C    &       =>   
 
Theoremvtoclb 2605* Implicit substitution of a class for a setvar variable. (Contributed by NM, 23-Dec-1993.)
 _V   &       &       &       =>   
 
Theoremvtoclgf 2606 Implicit substitution of a class for a setvar 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/_   &     F/   &       &       =>     V
 
Theoremvtoclg 2607* Implicit substitution of a class expression for a setvar variable. (Contributed by NM, 17-Apr-1995.)
   &       =>     V
 
Theoremvtoclbg 2608* Implicit substitution of a class for a setvar variable. (Contributed by NM, 29-Apr-1994.)
   &       &       =>     V
 
Theoremvtocl2gf 2609 Implicit substitution of a class for a setvar variable. (Contributed by NM, 25-Apr-1995.)
 F/_   &     F/_   &     F/_   &    
 F/   &     F/   &       &       &       =>     V  W
 
Theoremvtocl3gf 2610 Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
 F/_   &     F/_   &     F/_   &     F/_   &     F/_   &     F/_ C   &    
 F/   &     F/   &     F/   &       &       &     C    &       =>     V  W  C  X
 
Theoremvtocl2g 2611* Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 25-Apr-1995.)
   &       &       =>     V  W
 
Theoremvtoclgaf 2612* Implicit substitution of a class for a setvar variable. (Contributed by NM, 17-Feb-2006.) (Revised by Mario Carneiro, 10-Oct-2016.)
 F/_   &     F/   &       &       =>   
 
Theoremvtoclga 2613* Implicit substitution of a class for a setvar variable. (Contributed by NM, 20-Aug-1995.)
   &       =>   
 
Theoremvtocl2gaf 2614* Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 10-Aug-2013.)
 F/_   &     F/_   &     F/_   &    
 F/   &     F/   &       &       &     C  D    =>     C  D
 
Theoremvtocl2ga 2615* Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 20-Aug-1995.)
   &       &     C  D    =>     C  D
 
Theoremvtocl3gaf 2616* Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 11-Oct-2016.)
 F/_   &     F/_   &     F/_   &     F/_   &     F/_   &     F/_ C   &    
 F/   &     F/   &     F/   &       &       &     C    &     R  S  T    =>     R  S  C  T
 
Theoremvtocl3ga 2617* Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995.)
   &       &     C    &     D  R  S    =>     D  R  C  S
 
Theoremvtocleg 2618* Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Jan-2004.)
   =>     V
 
Theoremvtoclegft 2619* Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 2620.) (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)
 F/
 
Theoremvtoclef 2620* Implicit substitution of a class for a setvar variable. (Contributed by NM, 18-Aug-1993.)

 F/   &     _V   &       =>   
 
Theoremvtocle 2621* Implicit substitution of a class for a setvar variable. (Contributed by NM, 9-Sep-1993.)
 _V   &       =>   
 
Theoremvtoclri 2622* Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Nov-1994.)
   &       =>   
 
Theoremspcimgft 2623 A closed version of spcimgf 2627. (Contributed by Mario Carneiro, 4-Jan-2017.)

 F/   &     F/_   =>   
 
Theoremspcgft 2624 A closed version of spcgf 2629. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 4-Jan-2017.)

 F/   &     F/_   =>   
 
Theoremspcimegft 2625 A closed version of spcimegf 2628. (Contributed by Mario Carneiro, 4-Jan-2017.)

 F/   &     F/_   =>   
 
Theoremspcegft 2626 A closed version of spcegf 2630. (Contributed by Jim Kingdon, 22-Jun-2018.)

 F/   &     F/_   =>   
 
Theoremspcimgf 2627 Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by Mario Carneiro, 4-Jan-2017.)
 F/_   &     F/   &       =>     V
 
Theoremspcimegf 2628 Existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
 F/_   &     F/   &       =>     V
 
Theoremspcgf 2629 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/_   &     F/   &       =>     V
 
Theoremspcegf 2630 Existential specialization, using implicit substitution. (Contributed by NM, 2-Feb-1997.)
 F/_   &     F/   &       =>     V
 
Theoremspcimdv 2631* Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
   &       =>   
 
Theoremspcdv 2632* Rule of specialization, using implicit substitution. Analogous to rspcdv 2653. (Contributed by David Moews, 1-May-2017.)
   &       =>   
 
Theoremspcimedv 2633* Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
   &       =>   
 
Theoremspcgv 2634* Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.)
   =>     V
 
Theoremspcegv 2635* Existential specialization, using implicit substitution. (Contributed by NM, 14-Aug-1994.)
   =>     V
 
Theoremspc2egv 2636* Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
   =>     V  W
 
Theoremspc2gv 2637* Specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 27-Apr-2004.)
   =>     V  W
 
Theoremspc3egv 2638* Existential specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)
 C    =>     V  W  C  X
 
Theoremspc3gv 2639* Specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)
 C    =>     V  W  C  X
 
Theoremspcv 2640* Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.)
 _V   &       =>   
 
Theoremspcev 2641* Existential specialization, using implicit substitution. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)
 _V   &       =>   
 
Theoremspc2ev 2642* Existential specialization, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
 _V   &     _V   &       =>   
 
Theoremrspct 2643* A closed version of rspc 2644. (Contributed by Andrew Salmon, 6-Jun-2011.)

 F/   =>   
 
Theoremrspc 2644* Restricted specialization, using implicit substitution. (Contributed by NM, 19-Apr-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)

 F/   &       =>   
 
Theoremrspce 2645* Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) (Revised by Mario Carneiro, 11-Oct-2016.)

 F/   &       =>   
 
Theoremrspcv 2646* Restricted specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)
   =>   
 
Theoremrspccv 2647* Restricted specialization, using implicit substitution. (Contributed by NM, 2-Feb-2006.)
   =>   
 
Theoremrspcva 2648* Restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-2005.)
   =>   
 
Theoremrspccva 2649* Restricted specialization, using implicit substitution. (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
   =>   
 
Theoremrspcev 2650* Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)
   =>   
 
Theoremrspcimdv 2651* Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
   &       =>   
 
Theoremrspcimedv 2652* Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
   &       =>   
 
Theoremrspcdv 2653* Restricted specialization, using implicit substitution. (Contributed by NM, 17-Feb-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)
   &       =>   
 
Theoremrspcedv 2654* Restricted existential specialization, using implicit substitution. (Contributed by FL, 17-Apr-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)
   &       =>   
 
Theoremrspc2 2655* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 9-Nov-2012.)

 F/   &     F/   &       &       =>     C  D  C  D
 
Theoremrspc2v 2656* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-1999.)
   &       =>     C  D  C  D
 
Theoremrspc2va 2657* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 18-Jun-2014.)
   &       =>     C  D  C  D
 
Theoremrspc2ev 2658* 2-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 16-Oct-1999.)
   &       =>     C  D  C  D
 
Theoremrspc3v 2659* 3-variable restricted specialization, using implicit substitution. (Contributed by NM, 10-May-2005.)
   &       &     C    =>     R  S  C  T  R  S  T
 
Theoremrspc3ev 2660* 3-variable restricted existentional specialization, using implicit substitution. (Contributed by NM, 25-Jul-2012.)
   &       &     C    =>     R  S  C  T  R  S  T
 
Theoremeqvinc 2661* A variable introduction law for class equality. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 _V   =>   
 
Theoremeqvincg 2662* A variable introduction law for class equality, deduction version. (Contributed by Thierry Arnoux, 2-Mar-2017.)
 V
 
Theoremeqvincf 2663 A variable introduction law for class equality, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Sep-2003.)
 F/_   &     F/_   &     _V   =>   
 
Theoremalexeq 2664* Two ways to express substitution of for in . (Contributed by NM, 2-Mar-1995.)
 _V   =>   
 
Theoremceqex 2665* Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.)
 
Theoremceqsexg 2666* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 11-Oct-2004.)

 F/   &       =>     V
 
Theoremceqsexgv 2667* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996.)
   =>     V
 
Theoremceqsrexv 2668* Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 30-Apr-2004.)
   =>   
 
Theoremceqsrexbv 2669* Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by Mario Carneiro, 14-Mar-2014.)
   =>   
 
Theoremceqsrex2v 2670* Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 29-Oct-2005.)
   &       =>     C  D  C  D
 
Theoremclel2 2671* An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
 _V   =>   
 
Theoremclel3g 2672* An alternate definition of class membership when the class is a set. (Contributed by NM, 13-Aug-2005.)
 V
 
Theoremclel3 2673* An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
 _V   =>   
 
Theoremclel4 2674* An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
 _V   =>   
 
Theorempm13.183 2675* Compare theorem *13.183 in [WhiteheadRussell] p. 178. Only is required to be a set. (Contributed by Andrew Salmon, 3-Jun-2011.)
 V
 
Theoremrr19.3v 2676* Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 25-Oct-2012.)
 
Theoremrr19.28v 2677* Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 29-Oct-2012.)
 
Theoremelabgt 2678* Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 2682.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 {  |  }
 
Theoremelabgf 2679 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/_   &     F/   &       =>     {  |  }
 
Theoremelabf 2680* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 12-Oct-2016.)

 F/   &     _V   &       =>     {  |  }
 
Theoremelab 2681* Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.)
 _V   &       =>     {  |  }
 
Theoremelabg 2682* Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.)
   =>     V  {  |  }
 
Theoremelab2g 2683* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
   &     {  |  }   =>     V
 
Theoremelab2 2684* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
 _V   &       &     {  |  }   =>   
 
Theoremelab4g 2685* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.)
   &     {  |  }   =>     _V
 
Theoremelab3gf 2686 Membership in a class abstraction, with a weaker antecedent than elabgf 2679. (Contributed by NM, 6-Sep-2011.)
 F/_   &     F/   &       =>     {  |  }
 
Theoremelab3g 2687* Membership in a class abstraction, with a weaker antecedent than elabg 2682. (Contributed by NM, 29-Aug-2006.)
   =>     {  |  }
 
Theoremelab3 2688* Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.)
 _V   &       =>     {  |  }
 
Theoremelrabi 2689* Implication for the membership in a restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
 {  V  |  }  V
 
Theoremelrabf 2690 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/_   &     F/_   &     F/   &       =>     {  |  }
 
Theoremelrab3t 2691* Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 2693.) (Contributed by Thierry Arnoux, 31-Aug-2017.)
 {  | 
 }
 
Theoremelrab 2692* Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 21-May-1999.)
   =>     {  |  }
 
Theoremelrab3 2693* Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.)
   =>     {  |  }
 
Theoremelrab2 2694* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006.)
   &     C  {  |  }   =>     C
 
Theoremralab 2695* Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
   =>     {  |  }
 
Theoremralrab 2696* Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
   =>     {  | 
 }
 
Theoremrexab 2697* Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.)
   =>     {  |  }
 
Theoremrexrab 2698* Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.)
   =>     {  | 
 }
 
Theoremralab2 2699* Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
   =>     {  |  }
 
Theoremralrab2 2700* Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
   =>     {  | 
 }
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