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Theorem List for New Foundations Explorer - 2901-3000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgencbvex 2901* Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
   &       &       &       =>   
 
Theoremgencbvex2 2902* Restatement of gencbvex 2901 with weaker hypotheses. (Contributed by Jeffrey Hankins, 6-Dec-2006.)
   &       &       &       =>   
 
Theoremgencbval 2903* Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.)
   &       &       &       =>   
 
Theoremsbhypf 2904* Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3171. (Contributed by Raph Levien, 10-Apr-2004.)

 F/   &       =>   
 
Theoremvtoclgft 2905 Closed theorem form of vtoclgf 2913. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.)
 F/_  F/
 
Theoremvtocldf 2906 Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)
   &       &       &     F/   &     F/_   &     F/   =>   
 
Theoremvtocld 2907* Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)
   &       &       =>   
 
Theoremvtoclf 2908* Implicit substitution of a class for a setvar variable. This is a generalization of chvar 1986. (Contributed by NM, 30-Aug-1993.)

 F/   &       &       &       =>   
 
Theoremvtocl 2909* Implicit substitution of a class for a setvar variable. (Contributed by NM, 30-Aug-1993.)
   &       &       =>   
 
Theoremvtocl2 2910* Implicit substitution of classes for setvar variables. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
   &       &       &       =>   
 
Theoremvtocl3 2911* Implicit substitution of classes for setvar variables. (Contributed by NM, 3-Jun-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
   &       &       &       &       =>   
 
Theoremvtoclb 2912* Implicit substitution of a class for a setvar variable. (Contributed by NM, 23-Dec-1993.)
   &       &       &       =>   
 
Theoremvtoclgf 2913 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/   &       &       =>   
 
Theoremvtoclg 2914* Implicit substitution of a class expression for a setvar variable. (Contributed by NM, 17-Apr-1995.)
   &       =>   
 
Theoremvtoclbg 2915* Implicit substitution of a class for a setvar variable. (Contributed by NM, 29-Apr-1994.)
   &       &       =>   
 
Theoremvtocl2gf 2916 Implicit substitution of a class for a setvar variable. (Contributed by NM, 25-Apr-1995.)
 F/_   &     F/_   &     F/_   &    
 F/   &     F/   &       &       &       =>   
 
Theoremvtocl3gf 2917 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/_   &    
 F/   &     F/   &     F/   &       &       &       &       =>   
 
Theoremvtocl2g 2918* Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 25-Apr-1995.)
   &       &       =>   
 
Theoremvtoclgaf 2919* 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 2920* Implicit substitution of a class for a setvar variable. (Contributed by NM, 20-Aug-1995.)
   &       =>   
 
Theoremvtocl2gaf 2921* Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 10-Aug-2013.)
 F/_   &     F/_   &     F/_   &    
 F/   &     F/   &       &       &       =>   
 
Theoremvtocl2ga 2922* Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 20-Aug-1995.)
   &       &       =>   
 
Theoremvtocl3gaf 2923* 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/_   &    
 F/   &     F/   &     F/   &       &       &       &       =>   
 
Theoremvtocl3ga 2924* Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995.)
   &       &       &       =>   
 
Theoremvtocleg 2925* Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Jan-2004.)
   =>   
 
Theoremvtoclegft 2926* Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 2927.) (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)
 F/
 
Theoremvtoclef 2927* Implicit substitution of a class for a setvar variable. (Contributed by NM, 18-Aug-1993.)

 F/   &       &       =>   
 
Theoremvtocle 2928* Implicit substitution of a class for a setvar variable. (Contributed by NM, 9-Sep-1993.)
   &       =>   
 
Theoremvtoclri 2929* Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Nov-1994.)
   &       =>   
 
Theoremspcimgft 2930 A closed version of spcimgf 2932. (Contributed by Mario Carneiro, 4-Jan-2017.)

 F/   &     F/_   =>   
 
Theoremspcgft 2931 A closed version of spcgf 2934. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 4-Jan-2017.)

 F/   &     F/_   =>   
 
Theoremspcimgf 2932 Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by Mario Carneiro, 4-Jan-2017.)
 F/_   &     F/   &       =>   
 
Theoremspcimegf 2933 Existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
 F/_   &     F/   &       =>   
 
Theoremspcgf 2934 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/   &       =>   
 
Theoremspcegf 2935 Existential specialization, using implicit substitution. (Contributed by NM, 2-Feb-1997.)
 F/_   &     F/   &       =>   
 
Theoremspcimdv 2936* Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
   &       =>   
 
Theoremspcdv 2937* Rule of specialization, using implicit substitution. Analogous to rspcdv 2958. (Contributed by David Moews, 1-May-2017.)
   &       =>   
 
Theoremspcimedv 2938* Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
   &       =>   
 
Theoremspcgv 2939* Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.)
   =>   
 
Theoremspcegv 2940* Existential specialization, using implicit substitution. (Contributed by NM, 14-Aug-1994.)
   =>   
 
Theoremspc2egv 2941* Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
   =>   
 
Theoremspc2gv 2942* Specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 27-Apr-2004.)
   =>   
 
Theoremspc3egv 2943* Existential specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)
   =>   
 
Theoremspc3gv 2944* Specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)
   =>   
 
Theoremspcv 2945* Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.)
   &       =>   
 
Theoremspcev 2946* Existential specialization, using implicit substitution. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)
   &       =>   
 
Theoremspc2ev 2947* Existential specialization, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
   &       &       =>   
 
Theoremrspct 2948* A closed version of rspc 2949. (Contributed by Andrew Salmon, 6-Jun-2011.)

 F/   =>   
 
Theoremrspc 2949* Restricted specialization, using implicit substitution. (Contributed by NM, 19-Apr-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)

 F/   &       =>   
 
Theoremrspce 2950* Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) (Revised by Mario Carneiro, 11-Oct-2016.)

 F/   &       =>   
 
Theoremrspcv 2951* Restricted specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)
   =>   
 
Theoremrspccv 2952* Restricted specialization, using implicit substitution. (Contributed by NM, 2-Feb-2006.)
   =>   
 
Theoremrspcva 2953* Restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-2005.)
   =>   
 
Theoremrspccva 2954* Restricted specialization, using implicit substitution. (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
   =>   
 
Theoremrspcev 2955* Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)
   =>   
 
Theoremrspcimdv 2956* Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
   &       =>   
 
Theoremrspcimedv 2957* Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
   &       =>   
 
Theoremrspcdv 2958* Restricted specialization, using implicit substitution. (Contributed by NM, 17-Feb-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)
   &       =>   
 
Theoremrspcedv 2959* Restricted existential specialization, using implicit substitution. (Contributed by FL, 17-Apr-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)
   &       =>   
 
Theoremrspc2 2960* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 9-Nov-2012.)

 F/   &     F/   &       &       =>   
 
Theoremrspc2v 2961* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-1999.)
   &       =>   
 
Theoremrspc2va 2962* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 18-Jun-2014.)
   &       =>   
 
Theoremrspc2ev 2963* 2-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 16-Oct-1999.)
   &       =>   
 
Theoremrspc3v 2964* 3-variable restricted specialization, using implicit substitution. (Contributed by NM, 10-May-2005.)
   &       &       =>   
 
Theoremrspc3ev 2965* 3-variable restricted existentional specialization, using implicit substitution. (Contributed by NM, 25-Jul-2012.)
   &       &       =>   
 
Theoremeqvinc 2966* A variable introduction law for class equality. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
   =>   
 
Theoremeqvincf 2967 A variable introduction law for class equality, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Sep-2003.)
 F/_   &     F/_   &       =>   
 
Theoremalexeq 2968* Two ways to express substitution of for in . (Contributed by NM, 2-Mar-1995.)
   =>   
 
Theoremceqex 2969* Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.)
 
Theoremceqsexg 2970* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 11-Oct-2004.)

 F/   &       =>   
 
Theoremceqsexgv 2971* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996.)
   =>   
 
Theoremceqsrexv 2972* Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 30-Apr-2004.)
   =>   
 
Theoremceqsrexbv 2973* Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by Mario Carneiro, 14-Mar-2014.)
   =>   
 
Theoremceqsrex2v 2974* Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 29-Oct-2005.)
   &       =>   
 
Theoremclel2 2975* An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
   =>   
 
Theoremclel3g 2976* An alternate definition of class membership when the class is a set. (Contributed by NM, 13-Aug-2005.)
 
Theoremclel3 2977* An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
   =>   
 
Theoremclel4 2978* An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
   =>   
 
Theorempm13.183 2979* Compare theorem *13.183 in [WhiteheadRussell] p. 178. Only is required to be a set. (Contributed by Andrew Salmon, 3-Jun-2011.)
 
Theoremrr19.3v 2980* Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. We don't need the non-empty class condition of r19.3rzv 3643 when there is an outer quantifier. (Contributed by NM, 25-Oct-2012.)
 
Theoremrr19.28v 2981* Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. We don't need the non-empty class condition of r19.28zv 3645 when there is an outer quantifier. (Contributed by NM, 29-Oct-2012.)
 
Theoremelabgt 2982* Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 2986.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 
Theoremelabgf 2983 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 2984* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 12-Oct-2016.)

 F/   &       &       =>   
 
Theoremelab 2985* Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.)
   &       =>   
 
Theoremelabg 2986* Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.)
   =>   
 
Theoremelab2g 2987* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
   &       =>   
 
Theoremelab2 2988* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
   &       &       =>   
 
Theoremelab4g 2989* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.)
   &       =>   
 
Theoremelab3gf 2990 Membership in a class abstraction, with a weaker antecedent than elabgf 2983. (Contributed by NM, 6-Sep-2011.)
 F/_   &     F/   &       =>   
 
Theoremelab3g 2991* Membership in a class abstraction, with a weaker antecedent than elabg 2986. (Contributed by NM, 29-Aug-2006.)
   =>   
 
Theoremelab3 2992* Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.)
   &       =>   
 
Theoremelrabf 2993 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/   &       =>   
 
Theoremelrab 2994* Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 21-May-1999.)
   =>   
 
Theoremelrab3 2995* Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.)
   =>   
 
Theoremelrab2 2996* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006.)
   &       =>   
 
Theoremralab 2997* Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
   =>   
 
Theoremralrab 2998* Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
   =>   
 
Theoremrexab 2999* Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.)
   =>   
 
Theoremrexrab 3000* Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.)
   =>   
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