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Theorem List for Intuitionistic Logic Explorer - 2601-2700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremvtoclri 2601* Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Nov-1994.)
(x = A → (φψ))    &   x B φ       (A Bψ)

Theoremspcimgft 2602 A closed version of spcimgf 2606. (Contributed by Mario Carneiro, 4-Jan-2017.)
xψ    &   xA       (x(x = A → (φψ)) → (A B → (xφψ)))

Theoremspcgft 2603 A closed version of spcgf 2608. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 4-Jan-2017.)
xψ    &   xA       (x(x = A → (φψ)) → (A B → (xφψ)))

Theoremspcimegft 2604 A closed version of spcimegf 2607. (Contributed by Mario Carneiro, 4-Jan-2017.)
xψ    &   xA       (x(x = A → (ψφ)) → (A B → (ψxφ)))

Theoremspcegft 2605 A closed version of spcegf 2609. (Contributed by Jim Kingdon, 22-Jun-2018.)
xψ    &   xA       (x(x = A → (φψ)) → (A B → (ψxφ)))

Theoremspcimgf 2606 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 2607 Existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
xA    &   xψ    &   (x = A → (ψφ))       (A 𝑉 → (ψxφ))

Theoremspcgf 2608 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 2609 Existential specialization, using implicit substitution. (Contributed by NM, 2-Feb-1997.)
xA    &   xψ    &   (x = A → (φψ))       (A 𝑉 → (ψxφ))

Theoremspcimdv 2610* Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
(φA B)    &   ((φ x = A) → (ψχ))       (φ → (xψχ))

Theoremspcdv 2611* Rule of specialization, using implicit substitution. Analogous to rspcdv 2632. (Contributed by David Moews, 1-May-2017.)
(φA B)    &   ((φ x = A) → (ψχ))       (φ → (xψχ))

Theoremspcimedv 2612* Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
(φA B)    &   ((φ x = A) → (χψ))       (φ → (χxψ))

Theoremspcgv 2613* 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 2614* Existential specialization, using implicit substitution. (Contributed by NM, 14-Aug-1994.)
(x = A → (φψ))       (A 𝑉 → (ψxφ))

Theoremspc2egv 2615* 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 2616* Specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 27-Apr-2004.)
((x = A y = B) → (φψ))       ((A 𝑉 B 𝑊) → (xyφψ))

Theoremspc3egv 2617* 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 2618* 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 2619* Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.)
A V    &   (x = A → (φψ))       (xφψ)

Theoremspcev 2620* 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 2621* Existential specialization, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
A V    &   B V    &   ((x = A y = B) → (φψ))       (ψxyφ)

Theoremrspct 2622* A closed version of rspc 2623. (Contributed by Andrew Salmon, 6-Jun-2011.)
xψ       (x(x = A → (φψ)) → (A B → (x B φψ)))

Theoremrspc 2623* 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 2624* 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 2625* Restricted specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)
(x = A → (φψ))       (A B → (x B φψ))

Theoremrspccv 2626* Restricted specialization, using implicit substitution. (Contributed by NM, 2-Feb-2006.)
(x = A → (φψ))       (x B φ → (A Bψ))

Theoremrspcva 2627* Restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-2005.)
(x = A → (φψ))       ((A B x B φ) → ψ)

Theoremrspccva 2628* 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 2629* Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)
(x = A → (φψ))       ((A B ψ) → x B φ)

Theoremrspcimdv 2630* Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
(φA B)    &   ((φ x = A) → (ψχ))       (φ → (x B ψχ))

Theoremrspcimedv 2631* Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
(φA B)    &   ((φ x = A) → (χψ))       (φ → (χx B ψ))

Theoremrspcdv 2632* 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 2633* 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 2634* 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 2635* 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 2636* 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 2637* 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 2638* 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 2639* 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 2640* 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))

Theoremeqvincg 2641* A variable introduction law for class equality, deduction version. (Contributed by Thierry Arnoux, 2-Mar-2017.)
(A 𝑉 → (A = Bx(x = A x = B)))

Theoremeqvincf 2642 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 2643* 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 2644* Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.)
(x = A → (φx(x = A φ)))

Theoremceqsexg 2645* 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 2646* 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 2647* 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 2648* 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 2649* 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 2650* 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 2651* 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 2652* 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 2653* 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 2654* 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 2655* Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 25-Oct-2012.)
(x A y A φx A φ)

Theoremrr19.28v 2656* Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 29-Oct-2012.)
(x A y A (φ ψ) ↔ x A (φ y A ψ))

Theoremelabgt 2657* Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 2661.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
((A B x(x = A → (φψ))) → (A {xφ} ↔ ψ))

Theoremelabgf 2658 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 {xφ} ↔ ψ))

Theoremelabf 2659* 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 {xφ} ↔ ψ)

Theoremelab 2660* 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 {xφ} ↔ ψ)

Theoremelabg 2661* 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 {xφ} ↔ ψ))

Theoremelab2g 2662* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
(x = A → (φψ))    &   B = {xφ}       (A 𝑉 → (A Bψ))

Theoremelab2 2663* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
A V    &   (x = A → (φψ))    &   B = {xφ}       (A Bψ)

Theoremelab4g 2664* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.)
(x = A → (φψ))    &   B = {xφ}       (A B ↔ (A V ψ))

Theoremelab3gf 2665 Membership in a class abstraction, with a weaker antecedent than elabgf 2658. (Contributed by NM, 6-Sep-2011.)
xA    &   xψ    &   (x = A → (φψ))       ((ψA B) → (A {xφ} ↔ ψ))

Theoremelab3g 2666* Membership in a class abstraction, with a weaker antecedent than elabg 2661. (Contributed by NM, 29-Aug-2006.)
(x = A → (φψ))       ((ψA B) → (A {xφ} ↔ ψ))

Theoremelab3 2667* Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.)
(ψA V)    &   (x = A → (φψ))       (A {xφ} ↔ ψ)

Theoremelrabi 2668* Implication for the membership in a restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
(A {x 𝑉φ} → A 𝑉)

Theoremelrabf 2669 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 Bφ} ↔ (A B ψ))

Theoremelrab3t 2670* Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 2672.) (Contributed by Thierry Arnoux, 31-Aug-2017.)
((x(x = A → (φψ)) A B) → (A {x Bφ} ↔ ψ))

Theoremelrab 2671* Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 21-May-1999.)
(x = A → (φψ))       (A {x Bφ} ↔ (A B ψ))

Theoremelrab3 2672* Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.)
(x = A → (φψ))       (A B → (A {x Bφ} ↔ ψ))

Theoremelrab2 2673* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006.)
(x = A → (φψ))    &   𝐶 = {x Bφ}       (A 𝐶 ↔ (A B ψ))

Theoremralab 2674* Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
(y = x → (φψ))       (x {yφ}χx(ψχ))

Theoremralrab 2675* Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
(y = x → (φψ))       (x {y Aφ}χx A (ψχ))

Theoremrexab 2676* Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.)
(y = x → (φψ))       (x {yφ}χx(ψ χ))

Theoremrexrab 2677* 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 Aφ}χx A (ψ χ))

Theoremralab2 2678* Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
(x = y → (ψχ))       (x {yφ}ψy(φχ))

Theoremralrab2 2679* Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
(x = y → (ψχ))       (x {y Aφ}ψy A (φχ))

Theoremrexab2 2680* Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
(x = y → (ψχ))       (x {yφ}ψy(φ χ))

Theoremrexrab2 2681* Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
(x = y → (ψχ))       (x {y Aφ}ψy A (φ χ))

Theoremabidnf 2682* Identity used to create closed-form versions of bound-variable hypothesis builders for class expressions. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Mario Carneiro, 12-Oct-2016.)
(xA → {zx z A} = A)

Theoremdedhb 2683* A deduction theorem for converting the inference xA => φ into a closed theorem. Use nfa1 1412 and nfab 2160 to eliminate the hypothesis of the substitution instance ψ of the inference. For converting the inference form into a deduction form, abidnf 2682 is useful. (Contributed by NM, 8-Dec-2006.)
(A = {zx z A} → (φψ))    &   ψ       (xAφ)

Theoremeqeu 2684* A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.)
(x = A → (φψ))       ((A B ψ x(φx = A)) → ∃!xφ)

Theoremeueq 2685* Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.)
(A V ↔ ∃!x x = A)

Theoremeueq1 2686* Equality has existential uniqueness. (Contributed by NM, 5-Apr-1995.)
A V       ∃!x x = A

Theoremeueq2dc 2687* Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.)
A V    &   B V       (DECID φ∃!x((φ x = A) φ x = B)))

Theoremeueq3dc 2688* Equality has existential uniqueness (split into 3 cases). (Contributed by NM, 5-Apr-1995.) (Proof shortened by Mario Carneiro, 28-Sep-2015.)
A V    &   B V    &   𝐶 V    &    ¬ (φ ψ)       (DECID φ → (DECID ψ∃!x((φ x = A) (¬ (φ ψ) x = B) (ψ x = 𝐶))))

Theoremmoeq 2689* There is at most one set equal to a class. (Contributed by NM, 8-Mar-1995.)
∃*x x = A

Theoremmoeq3dc 2690* "At most one" property of equality (split into 3 cases). (Contributed by Jim Kingdon, 7-Jul-2018.)
A V    &   B V    &   𝐶 V    &    ¬ (φ ψ)       (DECID φ → (DECID ψ∃*x((φ x = A) (¬ (φ ψ) x = B) (ψ x = 𝐶))))

Theoremmosubt 2691* "At most one" remains true after substitution. (Contributed by Jim Kingdon, 18-Jan-2019.)
(y∃*xφ∃*xy(y = A φ))

Theoremmosub 2692* "At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995.)
∃*xφ       ∃*xy(y = A φ)

Theoremmo2icl 2693* Theorem for inferring "at most one." (Contributed by NM, 17-Oct-1996.)
(x(φx = A) → ∃*xφ)

Theoremmob2 2694* Consequence of "at most one." (Contributed by NM, 2-Jan-2015.)
(x = A → (φψ))       ((A B ∃*xφ φ) → (x = Aψ))

Theoremmoi2 2695* Consequence of "at most one." (Contributed by NM, 29-Jun-2008.)
(x = A → (φψ))       (((A B ∃*xφ) (φ ψ)) → x = A)

Theoremmob 2696* Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.)
(x = A → (φψ))    &   (x = B → (φχ))       (((A 𝐶 B 𝐷) ∃*xφ ψ) → (A = Bχ))

Theoremmoi 2697* Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.)
(x = A → (φψ))    &   (x = B → (φχ))       (((A 𝐶 B 𝐷) ∃*xφ (ψ χ)) → A = B)

Theoremmorex 2698* Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.)
B V    &   (x = B → (φψ))       ((x A φ ∃*xφ) → (ψB A))

Theoremeuxfr2dc 2699* Transfer existential uniqueness from a variable x to another variable y contained in expression A. (Contributed by NM, 14-Nov-2004.)
A V    &   ∃*y x = A       (DECID yx(x = A φ) → (∃!xy(x = A φ) ↔ ∃!yφ))

Theoremeuxfrdc 2700* Transfer existential uniqueness from a variable x to another variable y contained in expression A. (Contributed by NM, 14-Nov-2004.)
A V    &   ∃!y x = A    &   (x = A → (φψ))       (DECID yx(x = A ψ) → (∃!xφ∃!yψ))

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