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.)
A V    &   (A = y → (φψ))    &   (A = y → (χθ))    &   (θx(χ A = y))       (x(χ φ) ↔ y(θ ψ))

Theoremgencbvex2 2902* Restatement of gencbvex 2901 with weaker hypotheses. (Contributed by Jeffrey Hankins, 6-Dec-2006.)
A V    &   (A = y → (φψ))    &   (A = y → (χθ))    &   (θx(χ A = y))       (x(χ φ) ↔ y(θ ψ))

Theoremgencbval 2903* Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.)
A V    &   (A = y → (φψ))    &   (A = y → (χθ))    &   (θx(χ A = y))       (x(χφ) ↔ y(θψ))

Theoremsbhypf 2904* Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3171. (Contributed by Raph Levien, 10-Apr-2004.)
xψ    &   (x = A → (φψ))       (y = A → ([y / x]φψ))

Theoremvtoclgft 2905 Closed theorem form of vtoclgf 2913. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.)
(((xA xψ) (x(x = A → (φψ)) xφ) A V) → ψ)

Theoremvtocldf 2906 Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)
(φA V)    &   ((φ x = A) → (ψχ))    &   (φψ)    &   xφ    &   (φxA)    &   (φ → Ⅎxχ)       (φχ)

Theoremvtocld 2907* Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)
(φA V)    &   ((φ x = A) → (ψχ))    &   (φψ)       (φχ)

Theoremvtoclf 2908* Implicit substitution of a class for a setvar variable. This is a generalization of chvar 1986. (Contributed by NM, 30-Aug-1993.)
xψ    &   A V    &   (x = A → (φψ))    &   φ       ψ

Theoremvtocl 2909* Implicit substitution of a class for a setvar variable. (Contributed by NM, 30-Aug-1993.)
A V    &   (x = A → (φψ))    &   φ       ψ

Theoremvtocl2 2910* Implicit substitution of classes for setvar variables. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
A V    &   B V    &   ((x = A y = B) → (φψ))    &   φ       ψ

Theoremvtocl3 2911* Implicit substitution of classes for setvar variables. (Contributed by NM, 3-Jun-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
A V    &   B V    &   C V    &   ((x = A y = B z = C) → (φψ))    &   φ       ψ

Theoremvtoclb 2912* Implicit substitution of a class for a setvar variable. (Contributed by NM, 23-Dec-1993.)
A V    &   (x = A → (φχ))    &   (x = A → (ψθ))    &   (φψ)       (χθ)

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.)
xA    &   xψ    &   (x = A → (φψ))    &   φ       (A Vψ)

Theoremvtoclg 2914* Implicit substitution of a class expression for a setvar variable. (Contributed by NM, 17-Apr-1995.)
(x = A → (φψ))    &   φ       (A Vψ)

Theoremvtoclbg 2915* Implicit substitution of a class for a setvar variable. (Contributed by NM, 29-Apr-1994.)
(x = A → (φχ))    &   (x = A → (ψθ))    &   (φψ)       (A V → (χθ))

Theoremvtocl2gf 2916 Implicit substitution of a class for a setvar variable. (Contributed by NM, 25-Apr-1995.)
xA    &   yA    &   yB    &   xψ    &   yχ    &   (x = A → (φψ))    &   (y = B → (ψχ))    &   φ       ((A V B W) → χ)

Theoremvtocl3gf 2917 Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
xA    &   yA    &   zA    &   yB    &   zB    &   zC    &   xψ    &   yχ    &   zθ    &   (x = A → (φψ))    &   (y = B → (ψχ))    &   (z = C → (χθ))    &   φ       ((A V B W C X) → θ)

Theoremvtocl2g 2918* Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 25-Apr-1995.)
(x = A → (φψ))    &   (y = B → (ψχ))    &   φ       ((A V B W) → χ)

Theoremvtoclgaf 2919* Implicit substitution of a class for a setvar variable. (Contributed by NM, 17-Feb-2006.) (Revised by Mario Carneiro, 10-Oct-2016.)
xA    &   xψ    &   (x = A → (φψ))    &   (x Bφ)       (A Bψ)

Theoremvtoclga 2920* Implicit substitution of a class for a setvar variable. (Contributed by NM, 20-Aug-1995.)
(x = A → (φψ))    &   (x Bφ)       (A Bψ)

Theoremvtocl2gaf 2921* Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 10-Aug-2013.)
xA    &   yA    &   yB    &   xψ    &   yχ    &   (x = A → (φψ))    &   (y = B → (ψχ))    &   ((x C y D) → φ)       ((A C B D) → χ)

Theoremvtocl2ga 2922* Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 20-Aug-1995.)
(x = A → (φψ))    &   (y = B → (ψχ))    &   ((x C y D) → φ)       ((A C B D) → χ)

Theoremvtocl3gaf 2923* Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 11-Oct-2016.)
xA    &   yA    &   zA    &   yB    &   zB    &   zC    &   xψ    &   yχ    &   zθ    &   (x = A → (φψ))    &   (y = B → (ψχ))    &   (z = C → (χθ))    &   ((x R y S z T) → φ)       ((A R B S C T) → θ)

Theoremvtocl3ga 2924* Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995.)
(x = A → (φψ))    &   (y = B → (ψχ))    &   (z = C → (χθ))    &   ((x D y R z S) → φ)       ((A D B R C S) → θ)

Theoremvtocleg 2925* Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Jan-2004.)
(x = Aφ)       (A Vφ)

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.)
((A B xφ x(x = Aφ)) → φ)

Theoremvtoclef 2927* Implicit substitution of a class for a setvar variable. (Contributed by NM, 18-Aug-1993.)
xφ    &   A V    &   (x = Aφ)       φ

Theoremvtocle 2928* Implicit substitution of a class for a setvar variable. (Contributed by NM, 9-Sep-1993.)
A V    &   (x = Aφ)       φ

Theoremvtoclri 2929* Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Nov-1994.)
(x = A → (φψ))    &   x B φ       (A Bψ)

Theoremspcimgft 2930 A closed version of spcimgf 2932. (Contributed by Mario Carneiro, 4-Jan-2017.)
xψ    &   xA       (x(x = A → (φψ)) → (A B → (xφψ)))

Theoremspcgft 2931 A closed version of spcgf 2934. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 4-Jan-2017.)
xψ    &   xA       (x(x = A → (φψ)) → (A B → (xφψ)))

Theoremspcimgf 2932 Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by Mario Carneiro, 4-Jan-2017.)
xA    &   xψ    &   (x = A → (φψ))       (A V → (xφψ))

Theoremspcimegf 2933 Existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
xA    &   xψ    &   (x = A → (ψφ))       (A V → (ψxφ))

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.)
xA    &   xψ    &   (x = A → (φψ))       (A V → (xφψ))

Theoremspcegf 2935 Existential specialization, using implicit substitution. (Contributed by NM, 2-Feb-1997.)
xA    &   xψ    &   (x = A → (φψ))       (A V → (ψxφ))

Theoremspcimdv 2936* Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
(φA B)    &   ((φ x = A) → (ψχ))       (φ → (xψχ))

Theoremspcdv 2937* Rule of specialization, using implicit substitution. Analogous to rspcdv 2958. (Contributed by David Moews, 1-May-2017.)
(φA B)    &   ((φ x = A) → (ψχ))       (φ → (xψχ))

Theoremspcimedv 2938* Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
(φA B)    &   ((φ x = A) → (χψ))       (φ → (χxψ))

Theoremspcgv 2939* Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.)
(x = A → (φψ))       (A V → (xφψ))

Theoremspcegv 2940* Existential specialization, using implicit substitution. (Contributed by NM, 14-Aug-1994.)
(x = A → (φψ))       (A V → (ψxφ))

Theoremspc2egv 2941* Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
((x = A y = B) → (φψ))       ((A V B W) → (ψxyφ))

Theoremspc2gv 2942* Specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 27-Apr-2004.)
((x = A y = B) → (φψ))       ((A V B W) → (xyφψ))

Theoremspc3egv 2943* Existential specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)
((x = A y = B z = C) → (φψ))       ((A V B W C X) → (ψxyzφ))

Theoremspc3gv 2944* Specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)
((x = A y = B z = C) → (φψ))       ((A V B W C X) → (xyzφψ))

Theoremspcv 2945* Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.)
A V    &   (x = A → (φψ))       (xφψ)

Theoremspcev 2946* Existential specialization, using implicit substitution. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)
A V    &   (x = A → (φψ))       (ψxφ)

Theoremspc2ev 2947* Existential specialization, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
A V    &   B V    &   ((x = A y = B) → (φψ))       (ψxyφ)

Theoremrspct 2948* A closed version of rspc 2949. (Contributed by Andrew Salmon, 6-Jun-2011.)
xψ       (x(x = A → (φψ)) → (A B → (x B φψ)))

Theoremrspc 2949* Restricted specialization, using implicit substitution. (Contributed by NM, 19-Apr-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)
xψ    &   (x = A → (φψ))       (A B → (x B φψ))

Theoremrspce 2950* Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) (Revised by Mario Carneiro, 11-Oct-2016.)
xψ    &   (x = A → (φψ))       ((A B ψ) → x B φ)

Theoremrspcv 2951* Restricted specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)
(x = A → (φψ))       (A B → (x B φψ))

Theoremrspccv 2952* Restricted specialization, using implicit substitution. (Contributed by NM, 2-Feb-2006.)
(x = A → (φψ))       (x B φ → (A Bψ))

Theoremrspcva 2953* Restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-2005.)
(x = A → (φψ))       ((A B x B φ) → ψ)

Theoremrspccva 2954* Restricted specialization, using implicit substitution. (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
(x = A → (φψ))       ((x B φ A B) → ψ)

Theoremrspcev 2955* Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)
(x = A → (φψ))       ((A B ψ) → x B φ)

Theoremrspcimdv 2956* Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
(φA B)    &   ((φ x = A) → (ψχ))       (φ → (x B ψχ))

Theoremrspcimedv 2957* Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
(φA B)    &   ((φ x = A) → (χψ))       (φ → (χx B ψ))

Theoremrspcdv 2958* Restricted specialization, using implicit substitution. (Contributed by NM, 17-Feb-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)
(φA B)    &   ((φ x = A) → (ψχ))       (φ → (x B ψχ))

Theoremrspcedv 2959* Restricted existential specialization, using implicit substitution. (Contributed by FL, 17-Apr-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)
(φA B)    &   ((φ x = A) → (ψχ))       (φ → (χx B ψ))

Theoremrspc2 2960* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 9-Nov-2012.)
xχ    &   yψ    &   (x = A → (φχ))    &   (y = B → (χψ))       ((A C B D) → (x C y D φψ))

Theoremrspc2v 2961* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-1999.)
(x = A → (φχ))    &   (y = B → (χψ))       ((A C B D) → (x C y D φψ))

Theoremrspc2va 2962* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 18-Jun-2014.)
(x = A → (φχ))    &   (y = B → (χψ))       (((A C B D) x C y D φ) → ψ)

Theoremrspc2ev 2963* 2-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 16-Oct-1999.)
(x = A → (φχ))    &   (y = B → (χψ))       ((A C B D ψ) → x C y D φ)

Theoremrspc3v 2964* 3-variable restricted specialization, using implicit substitution. (Contributed by NM, 10-May-2005.)
(x = A → (φχ))    &   (y = B → (χθ))    &   (z = C → (θψ))       ((A R B S C T) → (x R y S z T φψ))

Theoremrspc3ev 2965* 3-variable restricted existentional specialization, using implicit substitution. (Contributed by NM, 25-Jul-2012.)
(x = A → (φχ))    &   (y = B → (χθ))    &   (z = C → (θψ))       (((A R B S C T) ψ) → x R y S z T φ)

Theoremeqvinc 2966* A variable introduction law for class equality. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
A V       (A = Bx(x = A x = B))

Theoremeqvincf 2967 A variable introduction law for class equality, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Sep-2003.)
xA    &   xB    &   A V       (A = Bx(x = A x = B))

Theoremalexeq 2968* Two ways to express substitution of A for x in φ. (Contributed by NM, 2-Mar-1995.)
A V       (x(x = Aφ) ↔ x(x = A φ))

Theoremceqex 2969* Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.)
(x = A → (φx(x = A φ)))

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.)
xψ    &   (x = A → (φψ))       (A V → (x(x = A φ) ↔ ψ))

Theoremceqsexgv 2971* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996.)
(x = A → (φψ))       (A V → (x(x = A φ) ↔ ψ))

Theoremceqsrexv 2972* Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 30-Apr-2004.)
(x = A → (φψ))       (A B → (x B (x = A φ) ↔ ψ))

Theoremceqsrexbv 2973* Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by Mario Carneiro, 14-Mar-2014.)
(x = A → (φψ))       (x B (x = A φ) ↔ (A B ψ))

Theoremceqsrex2v 2974* Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 29-Oct-2005.)
(x = A → (φψ))    &   (y = B → (ψχ))       ((A C B D) → (x C y D ((x = A y = B) φ) ↔ χ))

Theoremclel2 2975* An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
A V       (A Bx(x = Ax B))

Theoremclel3g 2976* An alternate definition of class membership when the class is a set. (Contributed by NM, 13-Aug-2005.)
(B V → (A Bx(x = B A x)))

Theoremclel3 2977* An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
B V       (A Bx(x = B A x))

Theoremclel4 2978* An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
B V       (A Bx(x = BA x))

Theorempm13.183 2979* Compare theorem *13.183 in [WhiteheadRussell] p. 178. Only A is required to be a set. (Contributed by Andrew Salmon, 3-Jun-2011.)
(A V → (A = Bz(z = Az = B)))

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.)
(x A y A φx A φ)

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.)
(x A y A (φ ψ) ↔ x A (φ y A ψ))

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.)
((A B x(x = A → (φψ))) → (A {x φ} ↔ ψ))

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.)
xA    &   xψ    &   (x = A → (φψ))       (A B → (A {x φ} ↔ ψ))

Theoremelabf 2984* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 12-Oct-2016.)
xψ    &   A V    &   (x = A → (φψ))       (A {x φ} ↔ ψ)

Theoremelab 2985* Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.)
A V    &   (x = A → (φψ))       (A {x φ} ↔ ψ)

Theoremelabg 2986* Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.)
(x = A → (φψ))       (A V → (A {x φ} ↔ ψ))

Theoremelab2g 2987* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
(x = A → (φψ))    &   B = {x φ}       (A V → (A Bψ))

Theoremelab2 2988* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
A V    &   (x = A → (φψ))    &   B = {x φ}       (A Bψ)

Theoremelab4g 2989* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.)
(x = A → (φψ))    &   B = {x φ}       (A B ↔ (A V ψ))

Theoremelab3gf 2990 Membership in a class abstraction, with a weaker antecedent than elabgf 2983. (Contributed by NM, 6-Sep-2011.)
xA    &   xψ    &   (x = A → (φψ))       ((ψA B) → (A {x φ} ↔ ψ))

Theoremelab3g 2991* Membership in a class abstraction, with a weaker antecedent than elabg 2986. (Contributed by NM, 29-Aug-2006.)
(x = A → (φψ))       ((ψA B) → (A {x φ} ↔ ψ))

Theoremelab3 2992* Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.)
(ψA V)    &   (x = A → (φψ))       (A {x φ} ↔ ψ)

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.)
xA    &   xB    &   xψ    &   (x = A → (φψ))       (A {x B φ} ↔ (A B ψ))

Theoremelrab 2994* Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 21-May-1999.)
(x = A → (φψ))       (A {x B φ} ↔ (A B ψ))

Theoremelrab3 2995* Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.)
(x = A → (φψ))       (A B → (A {x B φ} ↔ ψ))

Theoremelrab2 2996* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006.)
(x = A → (φψ))    &   C = {x B φ}       (A C ↔ (A B ψ))

Theoremralab 2997* Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
(y = x → (φψ))       (x {y φ}χx(ψχ))

Theoremralrab 2998* Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
(y = x → (φψ))       (x {y A φ}χx A (ψχ))

Theoremrexab 2999* Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.)
(y = x → (φψ))       (x {y φ}χx(ψ χ))

Theoremrexrab 3000* Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.)
(y = x → (φψ))       (x {y A φ}χx A (ψ χ))

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