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Theorem List for Intuitionistic Logic Explorer - 2501-2600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremreurmo 2501 Restricted existential uniqueness implies restricted "at most one." (Contributed by NM, 16-Jun-2017.)
(∃!x A φ∃*x A φ)
 
Theoremrmo5 2502 Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.)
(∃*x A φ ↔ (x A φ∃!x A φ))
 
Theoremnrexrmo 2503 Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.)
x A φ∃*x A φ)
 
Theoremcbvralf 2504 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 2505 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 2506* 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 2507* 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 2508* 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 2509* 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 2510* 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 2511* 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 2512* 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 2513* 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 2514* 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 2515* 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 2516* 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 2517* 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 2518* 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 2519* 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 2520* 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 2521* 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 2522* 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 2523* 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 2524 Equivalent wff's yield equal restricted class abstractions (inference rule). (Contributed by NM, 22-May-1999.)
(x A → (φψ))       {x Aφ} = {x Aψ}
 
Theoremrabbidva 2525* 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 2526* Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 10-Feb-1995.)
(φ → (ψχ))       (φ → {x Aψ} = {x Aχ})
 
Theoremrabeqf 2527 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 2528* Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.)
(A = B → {x Aφ} = {x Bφ})
 
Theoremrabeqbidv 2529* Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009.)
(φA = B)    &   (φ → (ψχ))       (φ → {x Aψ} = {x Bχ})
 
Theoremrabeqbidva 2530* Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.)
(φA = B)    &   ((φ x A) → (ψχ))       (φ → {x Aψ} = {x Bχ})
 
Theoremrabeq2i 2531 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 2532 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 2533* 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 2534 Extend class notation to include the universal class symbol.
class V
 
Theoremvjust 2535 Soundness justification theorem for df-v 2536. (Contributed by Rodolfo Medina, 27-Apr-2010.)
{xx = x} = {yy = y}
 
Definitiondf-v 2536 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 2537 All setvar variables are sets (see isset 2538). Theorem 6.8 of [Quine] p. 43. (Contributed by NM, 5-Aug-1993.)
x V
 
Theoremisset 2538* Two ways to say "A is a set": A class A is a member of the universal class V (see df-v 2536) 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 2019 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 2539 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 2540* A way to say "A is a set" (inference rule). (Contributed by NM, 5-Aug-1993.)
A V       x x = A
 
Theoremissetri 2541* A way to say "A is a set" (inference rule). (Contributed by NM, 5-Aug-1993.)
x x = A       A V
 
Theoremelex 2542 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 2543 If a class is a member of another class, it is a set. (Contributed by NM, 11-Jun-1994.)
A B       A V
 
Theoremelisset 2544* An element of a class exists. (Contributed by NM, 1-May-1995.)
(A 𝑉x x = A)
 
Theoremelex22 2545* 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 2546* If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.)
(A Bx x B)
 
Theoremralv 2547 A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
(x V φxφ)
 
Theoremrexv 2548 An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
(x V φxφ)
 
Theoremreuv 2549 A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.)
(∃!x V φ∃!xφ)
 
Theoremrmov 2550 A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(∃*x V φ∃*xφ)
 
Theoremrabab 2551 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 2552* 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 2553* 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 2554* Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
(xy A (φ ψ) ↔ y A (φ xψ))
 
Theoremrexcom4b 2555* Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
B V       (xy A (φ x = B) ↔ y A φ)
 
Theoremceqsalt 2556* Closed theorem version of ceqsalg 2558. (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 2557* Restricted quantifier version of ceqsalt 2556. (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 2558* 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 2559* 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 2560* 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 2561* Restricted quantifier version of ceqsalv 2560. (Contributed by NM, 21-Jun-2013.)
(x = A → (φψ))       (A B → (x B (x = Aφ) ↔ ψ))
 
Theoremgencl 2562* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
(θx(χ A = B))    &   (A = B → (φψ))    &   (χφ)       (θψ)
 
Theorem2gencl 2563* 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 2564* 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 2565* Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.)
(x = Aχ)    &   (χ → (φψ))       (A 𝑉 → (x(χ φ) ↔ ψ))
 
Theoremcgsex2g 2566* 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 2567* 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 2568* 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 2569* 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 2570* 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 2571* 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 2572* 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 2573* 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 2574* 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 2575* 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 2576* 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 2577* Restatement of gencbvex 2576 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 2578* 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 2579* 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 2580 Closed theorem form of vtoclgf 2588. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.)
(((xA xψ) (x(x = A → (φψ)) xφ) A 𝑉) → ψ)
 
Theoremvtocldf 2581 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 2582* 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 2583* Implicit substitution of a class for a setvar variable. This is a generalization of chvar 1623. (Contributed by NM, 30-Aug-1993.)
xψ    &   A V    &   (x = A → (φψ))    &   φ       ψ
 
Theoremvtocl 2584* Implicit substitution of a class for a setvar variable. (Contributed by NM, 30-Aug-1993.)
A V    &   (x = A → (φψ))    &   φ       ψ
 
Theoremvtocl2 2585* 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 2586* 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 2587* 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 2588 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 2589* Implicit substitution of a class expression for a setvar variable. (Contributed by NM, 17-Apr-1995.)
(x = A → (φψ))    &   φ       (A 𝑉ψ)
 
Theoremvtoclbg 2590* 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 2591 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 2592 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 2593* 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 2594* 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 2595* Implicit substitution of a class for a setvar variable. (Contributed by NM, 20-Aug-1995.)
(x = A → (φψ))    &   (x Bφ)       (A Bψ)
 
Theoremvtocl2gaf 2596* 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 2597* 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 2598* 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 2599* 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 2600* Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Jan-2004.)
(x = Aφ)       (A 𝑉φ)
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