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Theorem List for Intuitionistic Logic Explorer - 2501-2600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremreueq1 2501* Equality theorem for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)
(A = B → (∃!x A φ∃!x B φ))

Theoremrmoeq1 2502* Equality theorem for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(A = B → (∃*x A φ∃*x B φ))

Theoremraleqi 2503* Equality inference for restricted universal qualifier. (Contributed by Paul Chapman, 22-Jun-2011.)
A = B       (x A φx B φ)

Theoremrexeqi 2504* Equality inference for restricted existential qualifier. (Contributed by Mario Carneiro, 23-Apr-2015.)
A = B       (x A φx B φ)

Theoremraleqdv 2505* Equality deduction for restricted universal quantifier. (Contributed by NM, 13-Nov-2005.)
(φA = B)       (φ → (x A ψx B ψ))

Theoremrexeqdv 2506* Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.)
(φA = B)       (φ → (x A ψx B ψ))

Theoremraleqbi1dv 2507* Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
(A = B → (φψ))       (A = B → (x A φx B ψ))

Theoremrexeqbi1dv 2508* Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.)
(A = B → (φψ))       (A = B → (x A φx B ψ))

Theoremreueqd 2509* Equality deduction for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)
(A = B → (φψ))       (A = B → (∃!x A φ∃!x B ψ))

Theoremrmoeqd 2510* Equality deduction for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(A = B → (φψ))       (A = B → (∃*x A φ∃*x B ψ))

Theoremraleqbidv 2511* Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
(φA = B)    &   (φ → (ψχ))       (φ → (x A ψx B χ))

Theoremrexeqbidv 2512* Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
(φA = B)    &   (φ → (ψχ))       (φ → (x A ψx B χ))

Theoremraleqbidva 2513* Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
(φA = B)    &   ((φ x A) → (ψχ))       (φ → (x A ψx B χ))

Theoremrexeqbidva 2514* Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
(φA = B)    &   ((φ x A) → (ψχ))       (φ → (x A ψx B χ))

Theoremmormo 2515 Unrestricted "at most one" implies restricted "at most one". (Contributed by NM, 16-Jun-2017.)
(∃*xφ∃*x A φ)

Theoremreu5 2516 Restricted uniqueness in terms of "at most one." (Contributed by NM, 23-May-1999.) (Revised by NM, 16-Jun-2017.)
(∃!x A φ ↔ (x A φ ∃*x A φ))

Theoremreurex 2517 Restricted unique existence implies restricted existence. (Contributed by NM, 19-Aug-1999.)
(∃!x A φx A φ)

Theoremreurmo 2518 Restricted existential uniqueness implies restricted "at most one." (Contributed by NM, 16-Jun-2017.)
(∃!x A φ∃*x A φ)

Theoremrmo5 2519 Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.)
(∃*x A φ ↔ (x A φ∃!x A φ))

Theoremnrexrmo 2520 Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.)
x A φ∃*x A φ)

Theoremcbvralf 2521 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro, 9-Oct-2016.)
xA    &   yA    &   yφ    &   xψ    &   (x = y → (φψ))       (x A φy A ψ)

Theoremcbvrexf 2522 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.)
xA    &   yA    &   yφ    &   xψ    &   (x = y → (φψ))       (x A φy A ψ)

Theoremcbvral 2523* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.)
yφ    &   xψ    &   (x = y → (φψ))       (x A φy A ψ)

Theoremcbvrex 2524* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
yφ    &   xψ    &   (x = y → (φψ))       (x A φy A ψ)

Theoremcbvreu 2525* Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Mario Carneiro, 15-Oct-2016.)
yφ    &   xψ    &   (x = y → (φψ))       (∃!x A φ∃!y A ψ)

Theoremcbvrmo 2526* Change the bound variable of restricted "at most one" using implicit substitution. (Contributed by NM, 16-Jun-2017.)
yφ    &   xψ    &   (x = y → (φψ))       (∃*x A φ∃*y A ψ)

Theoremcbvralv 2527* Change the bound variable of a restricted universal quantifier using implicit substitution. (Contributed by NM, 28-Jan-1997.)
(x = y → (φψ))       (x A φy A ψ)

Theoremcbvrexv 2528* Change the bound variable of a restricted existential quantifier using implicit substitution. (Contributed by NM, 2-Jun-1998.)
(x = y → (φψ))       (x A φy A ψ)

Theoremcbvreuv 2529* Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
(x = y → (φψ))       (∃!x A φ∃!y A ψ)

Theoremcbvrmov 2530* Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(x = y → (φψ))       (∃*x A φ∃*y A ψ)

Theoremcbvraldva2 2531* 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.)
((φ x = y) → (ψχ))    &   ((φ x = y) → A = B)       (φ → (x A ψy B χ))

Theoremcbvrexdva2 2532* 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.)
((φ x = y) → (ψχ))    &   ((φ x = y) → A = B)       (φ → (x A ψy B χ))

Theoremcbvraldva 2533* Rule used to change the bound variable in a restricted universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
((φ x = y) → (ψχ))       (φ → (x A ψy A χ))

Theoremcbvrexdva 2534* Rule used to change the bound variable in a restricted existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
((φ x = y) → (ψχ))       (φ → (x A ψy A χ))

Theoremcbvral2v 2535* Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by NM, 10-Aug-2004.)
(x = z → (φχ))    &   (y = w → (χψ))       (x A y B φz A w B ψ)

Theoremcbvrex2v 2536* Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by FL, 2-Jul-2012.)
(x = z → (φχ))    &   (y = w → (χψ))       (x A y B φz A w B ψ)

Theoremcbvral3v 2537* Change bound variables of triple restricted universal quantification, using implicit substitution. (Contributed by NM, 10-May-2005.)
(x = w → (φχ))    &   (y = v → (χθ))    &   (z = u → (θψ))       (x A y B z 𝐶 φw A v B u 𝐶 ψ)

Theoremcbvralsv 2538* Change bound variable by using a substitution. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.)
(x A φy A [y / x]φ)

Theoremcbvrexsv 2539* Change bound variable by using a substitution. (Contributed by NM, 2-Mar-2008.) (Revised by Andrew Salmon, 11-Jul-2011.)
(x A φy A [y / x]φ)

Theoremsbralie 2540* Implicit to explicit substitution that swaps variables in a quantified expression. (Contributed by NM, 5-Sep-2004.)
(x = y → (φψ))       ([x / y]x y φy x ψ)

Theoremrabbiia 2541 Equivalent wff's yield equal restricted class abstractions (inference rule). (Contributed by NM, 22-May-1999.)
(x A → (φψ))       {x Aφ} = {x Aψ}

Theoremrabbidva 2542* Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 28-Nov-2003.)
((φ x A) → (ψχ))       (φ → {x Aψ} = {x Aχ})

Theoremrabbidv 2543* Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 10-Feb-1995.)
(φ → (ψχ))       (φ → {x Aψ} = {x Aχ})

Theoremrabeqf 2544 Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.)
xA    &   xB       (A = B → {x Aφ} = {x Bφ})

Theoremrabeq 2545* Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.)
(A = B → {x Aφ} = {x Bφ})

Theoremrabeqbidv 2546* Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009.)
(φA = B)    &   (φ → (ψχ))       (φ → {x Aψ} = {x Bχ})

Theoremrabeqbidva 2547* Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.)
(φA = B)    &   ((φ x A) → (ψχ))       (φ → {x Aψ} = {x Bχ})

Theoremrabeq2i 2548 Inference rule from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.)
A = {x Bφ}       (x A ↔ (x B φ))

Theoremcbvrab 2549 Rule to change the bound variable in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 9-Oct-2016.)
xA    &   yA    &   yφ    &   xψ    &   (x = y → (φψ))       {x Aφ} = {y Aψ}

Theoremcbvrabv 2550* Rule to change the bound variable in a restricted class abstraction, using implicit substitution. (Contributed by NM, 26-May-1999.)
(x = y → (φψ))       {x Aφ} = {y Aψ}

2.1.6  The universal class

Syntaxcvv 2551 Extend class notation to include the universal class symbol.
class V

Theoremvjust 2552 Soundness justification theorem for df-v 2553. (Contributed by Rodolfo Medina, 27-Apr-2010.)
{xx = x} = {yy = y}

Definitiondf-v 2553 Define the universal class. Definition 5.20 of [TakeutiZaring] p. 21. Also Definition 2.9 of [Quine] p. 19. (Contributed by NM, 5-Aug-1993.)
V = {xx = x}

Theoremvex 2554 All setvar variables are sets (see isset 2555). Theorem 6.8 of [Quine] p. 43. (Contributed by NM, 5-Aug-1993.)
x V

Theoremisset 2555* Two ways to say "A is a set": A class A is a member of the universal class V (see df-v 2553) if and only if the class A exists (i.e. there exists some set x equal to class A). Theorem 6.9 of [Quine] p. 43. Notational convention: We will use the notational device "A V " to mean "A is a set" very frequently, for example in uniex . Note the when A is not a set, it is called a proper class. In some theorems, such as uniexg , in order to shorten certain proofs we use the more general antecedent A 𝑉 instead of A V to mean "A is a set."

Note that a constant is implicitly considered distinct from all variables. This is why V is not included in the distinct variable list, even though df-clel 2033 requires that the expression substituted for B not contain x. (Also, the Metamath spec does not allow constants in the distinct variable list.) (Contributed by NM, 5-Aug-1993.)

(A V ↔ x x = A)

Theoremissetf 2556 A version of isset that does not require x and A to be distinct. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 10-Oct-2016.)
xA       (A V ↔ x x = A)

Theoremisseti 2557* A way to say "A is a set" (inference rule). (Contributed by NM, 5-Aug-1993.)
A V       x x = A

Theoremissetri 2558* A way to say "A is a set" (inference rule). (Contributed by NM, 5-Aug-1993.)
x x = A       A V

Theoremeqvisset 2559 A class equal to a variable is a set. Note the absence of dv condition, contrary to isset 2555 and issetri 2558. (Contributed by BJ, 27-Apr-2019.)
(x = AA V)

Theoremelex 2560 If a class is a member of another class, it is a set. Theorem 6.12 of [Quine] p. 44. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
(A BA V)

Theoremelexi 2561 If a class is a member of another class, it is a set. (Contributed by NM, 11-Jun-1994.)
A B       A V

Theoremelisset 2562* An element of a class exists. (Contributed by NM, 1-May-1995.)
(A 𝑉x x = A)

Theoremelex22 2563* If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.)
((A B A 𝐶) → x(x B x 𝐶))

Theoremelex2 2564* If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.)
(A Bx x B)

Theoremralv 2565 A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
(x V φxφ)

Theoremrexv 2566 An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
(x V φxφ)

Theoremreuv 2567 A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.)
(∃!x V φ∃!xφ)

Theoremrmov 2568 A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(∃*x V φ∃*xφ)

Theoremrabab 2569 A class abstraction restricted to the universe is unrestricted. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
{x V ∣ φ} = {xφ}

Theoremralcom4 2570* Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
(x A yφyx A φ)

Theoremrexcom4 2571* Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
(x A yφyx A φ)

Theoremrexcom4a 2572* Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
(xy A (φ ψ) ↔ y A (φ xψ))

Theoremrexcom4b 2573* Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
B V       (xy A (φ x = B) ↔ y A φ)

Theoremceqsalt 2574* Closed theorem version of ceqsalg 2576. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
((Ⅎxψ x(x = A → (φψ)) A 𝑉) → (x(x = Aφ) ↔ ψ))

Theoremceqsralt 2575* Restricted quantifier version of ceqsalt 2574. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
((Ⅎxψ x(x = A → (φψ)) A B) → (x B (x = Aφ) ↔ ψ))

Theoremceqsalg 2576* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 29-Oct-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
xψ    &   (x = A → (φψ))       (A 𝑉 → (x(x = Aφ) ↔ ψ))

Theoremceqsal 2577* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)
xψ    &   A V    &   (x = A → (φψ))       (x(x = Aφ) ↔ ψ)

Theoremceqsalv 2578* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)
A V    &   (x = A → (φψ))       (x(x = Aφ) ↔ ψ)

Theoremceqsralv 2579* Restricted quantifier version of ceqsalv 2578. (Contributed by NM, 21-Jun-2013.)
(x = A → (φψ))       (A B → (x B (x = Aφ) ↔ ψ))

Theoremgencl 2580* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
(θx(χ A = B))    &   (A = B → (φψ))    &   (χφ)       (θψ)

Theorem2gencl 2581* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
(𝐶 𝑆x 𝑅 A = 𝐶)    &   (𝐷 𝑆y 𝑅 B = 𝐷)    &   (A = 𝐶 → (φψ))    &   (B = 𝐷 → (ψχ))    &   ((x 𝑅 y 𝑅) → φ)       ((𝐶 𝑆 𝐷 𝑆) → χ)

Theorem3gencl 2582* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
(𝐷 𝑆x 𝑅 A = 𝐷)    &   (𝐹 𝑆y 𝑅 B = 𝐹)    &   (𝐺 𝑆z 𝑅 𝐶 = 𝐺)    &   (A = 𝐷 → (φψ))    &   (B = 𝐹 → (ψχ))    &   (𝐶 = 𝐺 → (χθ))    &   ((x 𝑅 y 𝑅 z 𝑅) → φ)       ((𝐷 𝑆 𝐹 𝑆 𝐺 𝑆) → θ)

Theoremcgsexg 2583* Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.)
(x = Aχ)    &   (χ → (φψ))       (A 𝑉 → (x(χ φ) ↔ ψ))

Theoremcgsex2g 2584* Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.)
((x = A y = B) → χ)    &   (χ → (φψ))       ((A 𝑉 B 𝑊) → (xy(χ φ) ↔ ψ))

Theoremcgsex4g 2585* An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.)
(((x = A y = B) (z = 𝐶 w = 𝐷)) → χ)    &   (χ → (φψ))       (((A 𝑅 B 𝑆) (𝐶 𝑅 𝐷 𝑆)) → (xyzw(χ φ) ↔ ψ))

Theoremceqsex 2586* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.)
xψ    &   A V    &   (x = A → (φψ))       (x(x = A φ) ↔ ψ)

Theoremceqsexv 2587* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.)
A V    &   (x = A → (φψ))       (x(x = A φ) ↔ ψ)

Theoremceqsex2 2588* Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)
xψ    &   yχ    &   A V    &   B V    &   (x = A → (φψ))    &   (y = B → (ψχ))       (xy(x = A y = B φ) ↔ χ)

Theoremceqsex2v 2589* Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)
A V    &   B V    &   (x = A → (φψ))    &   (y = B → (ψχ))       (xy(x = A y = B φ) ↔ χ)

Theoremceqsex3v 2590* Elimination of three existential quantifiers, using implicit substitution. (Contributed by NM, 16-Aug-2011.)
A V    &   B V    &   𝐶 V    &   (x = A → (φψ))    &   (y = B → (ψχ))    &   (z = 𝐶 → (χθ))       (xyz((x = A y = B z = 𝐶) φ) ↔ θ)

Theoremceqsex4v 2591* Elimination of four existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)
A V    &   B V    &   𝐶 V    &   𝐷 V    &   (x = A → (φψ))    &   (y = B → (ψχ))    &   (z = 𝐶 → (χθ))    &   (w = 𝐷 → (θτ))       (xyzw((x = A y = B) (z = 𝐶 w = 𝐷) φ) ↔ τ)

Theoremceqsex6v 2592* Elimination of six existential quantifiers, using implicit substitution. (Contributed by NM, 21-Sep-2011.)
A V    &   B V    &   𝐶 V    &   𝐷 V    &   𝐸 V    &   𝐹 V    &   (x = A → (φψ))    &   (y = B → (ψχ))    &   (z = 𝐶 → (χθ))    &   (w = 𝐷 → (θτ))    &   (v = 𝐸 → (τη))    &   (u = 𝐹 → (ηζ))       (xyzwvu((x = A y = B z = 𝐶) (w = 𝐷 v = 𝐸 u = 𝐹) φ) ↔ ζ)

Theoremceqsex8v 2593* Elimination of eight existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)
A V    &   B V    &   𝐶 V    &   𝐷 V    &   𝐸 V    &   𝐹 V    &   𝐺 V    &   𝐻 V    &   (x = A → (φψ))    &   (y = B → (ψχ))    &   (z = 𝐶 → (χθ))    &   (w = 𝐷 → (θτ))    &   (v = 𝐸 → (τη))    &   (u = 𝐹 → (ηζ))    &   (𝑡 = 𝐺 → (ζσ))    &   (𝑠 = 𝐻 → (σρ))       (xyzwvu𝑡𝑠(((x = A y = B) (z = 𝐶 w = 𝐷)) ((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻)) φ) ↔ ρ)

Theoremgencbvex 2594* 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 2595* Restatement of gencbvex 2594 with weaker hypotheses. (Contributed by Jeff Hankins, 6-Dec-2006.)
A V    &   (A = y → (φψ))    &   (A = y → (χθ))    &   (θx(χ A = y))       (x(χ φ) ↔ y(θ ψ))

Theoremgencbval 2596* Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof rewritten by Jim Kingdon, 20-Jun-2018.)
A V    &   (A = y → (φψ))    &   (A = y → (χθ))    &   (θx(χ A = y))       (x(χφ) ↔ y(θψ))

Theoremsbhypf 2597* Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf . (Contributed by Raph Levien, 10-Apr-2004.)
xψ    &   (x = A → (φψ))       (y = A → ([y / x]φψ))

Theoremvtoclgft 2598 Closed theorem form of vtoclgf 2606. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.)
(((xA xψ) (x(x = A → (φψ)) xφ) A 𝑉) → ψ)

Theoremvtocldf 2599 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 2600* Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)
(φA 𝑉)    &   ((φ x = A) → (ψχ))    &   (φψ)       (φχ)

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