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Theorem List for Intuitionistic Logic Explorer - 2601-2700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrabbidva2 2601* Equivalent wff's yield equal restricted class abstractions. (Contributed by Thierry Arnoux, 4-Feb-2017.)
(𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
 
Theoremrabbidva 2602* Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 28-Nov-2003.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
 
Theoremrabbidv 2603* Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 10-Feb-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
 
Theoremrabeqf 2604 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 → {𝑥𝐴𝜑} = {𝑥𝐵𝜑})
 
Theoremrabeqif 2605 Equality theorem for restricted class abstractions. Inference form of rabeqf 2604. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐴    &   𝑥𝐵    &   𝐴 = 𝐵       {𝑥𝐴𝜑} = {𝑥𝐵𝜑}
 
Theoremrabeq 2606* Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.)
(𝐴 = 𝐵 → {𝑥𝐴𝜑} = {𝑥𝐵𝜑})
 
Theoremrabeqi 2607* Equality theorem for restricted class abstractions. Inference form of rabeq 2606. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐴 = 𝐵       {𝑥𝐴𝜑} = {𝑥𝐵𝜑}
 
Theoremrabeqdv 2608* Equality of restricted class abstractions. Deduction form of rabeq 2606. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜓})
 
Theoremrabeqbidv 2609* Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
 
Theoremrabeqbidva 2610* Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
 
Theoremrabeq2i 2611 Inference rule from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.)
𝐴 = {𝑥𝐵𝜑}       (𝑥𝐴 ↔ (𝑥𝐵𝜑))
 
Theoremcbvrab 2612 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))       {𝑥𝐴𝜑} = {𝑦𝐴𝜓}
 
Theoremcbvrabv 2613* Rule to change the bound variable in a restricted class abstraction, using implicit substitution. (Contributed by NM, 26-May-1999.)
(𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝐴𝜑} = {𝑦𝐴𝜓}
 
2.1.6  The universal class
 
Syntaxcvv 2614 Extend class notation to include the universal class symbol.
class V
 
Theoremvjust 2615 Soundness justification theorem for df-v 2616. (Contributed by Rodolfo Medina, 27-Apr-2010.)
{𝑥𝑥 = 𝑥} = {𝑦𝑦 = 𝑦}
 
Definitiondf-v 2616 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 2617 All setvar variables are sets (see isset 2618). Theorem 6.8 of [Quine] p. 43. (Contributed by NM, 5-Aug-1993.)
𝑥 ∈ V
 
Theoremisset 2618* Two ways to say "𝐴 is a set": A class 𝐴 is a member of the universal class V (see df-v 2616) 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 4231. Note the when 𝐴 is not a set, it is called a proper class. In some theorems, such as uniexg 4232, in order to shorten certain proofs we use the more general antecedent 𝐴𝑉 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 2081 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, 26-May-1993.)

(𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
 
Theoremissetf 2619 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 2620* A way to say "𝐴 is a set" (inference rule). (Contributed by NM, 5-Aug-1993.)
𝐴 ∈ V       𝑥 𝑥 = 𝐴
 
Theoremissetri 2621* A way to say "𝐴 is a set" (inference rule). (Contributed by NM, 5-Aug-1993.)
𝑥 𝑥 = 𝐴       𝐴 ∈ V
 
Theoremeqvisset 2622 A class equal to a variable is a set. Note the absence of dv condition, contrary to isset 2618 and issetri 2621. (Contributed by BJ, 27-Apr-2019.)
(𝑥 = 𝐴𝐴 ∈ V)
 
Theoremelex 2623 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.)
(𝐴𝐵𝐴 ∈ V)
 
Theoremelexi 2624 If a class is a member of another class, it is a set. (Contributed by NM, 11-Jun-1994.)
𝐴𝐵       𝐴 ∈ V
 
Theoremelexd 2625 If a class is a member of another class, it is a set. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴𝑉)       (𝜑𝐴 ∈ V)
 
Theoremelisset 2626* An element of a class exists. (Contributed by NM, 1-May-1995.)
(𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
 
Theoremelex22 2627* If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.)
((𝐴𝐵𝐴𝐶) → ∃𝑥(𝑥𝐵𝑥𝐶))
 
Theoremelex2 2628* If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.)
(𝐴𝐵 → ∃𝑥 𝑥𝐵)
 
Theoremralv 2629 A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
(∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑)
 
Theoremrexv 2630 An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
(∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)
 
Theoremreuv 2631 A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.)
(∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑)
 
Theoremrmov 2632 A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥𝜑)
 
Theoremrabab 2633 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 2634* Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
(∀𝑥𝐴𝑦𝜑 ↔ ∀𝑦𝑥𝐴 𝜑)
 
Theoremrexcom4 2635* Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
(∃𝑥𝐴𝑦𝜑 ↔ ∃𝑦𝑥𝐴 𝜑)
 
Theoremrexcom4a 2636* Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
(∃𝑥𝑦𝐴 (𝜑𝜓) ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥𝜓))
 
Theoremrexcom4b 2637* Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
𝐵 ∈ V       (∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵) ↔ ∃𝑦𝐴 𝜑)
 
Theoremceqsalt 2638* Closed theorem version of ceqsalg 2640. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
 
Theoremceqsralt 2639* Restricted quantifier version of ceqsalt 2638. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
 
Theoremceqsalg 2640* 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))       (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
 
Theoremceqsal 2641* 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 2642* 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 2643* Restricted quantifier version of ceqsalv 2642. (Contributed by NM, 21-Jun-2013.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝐵 → (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
 
Theoremgencl 2644* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
(𝜃 ↔ ∃𝑥(𝜒𝐴 = 𝐵))    &   (𝐴 = 𝐵 → (𝜑𝜓))    &   (𝜒𝜑)       (𝜃𝜓)
 
Theorem2gencl 2645* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
(𝐶𝑆 ↔ ∃𝑥𝑅 𝐴 = 𝐶)    &   (𝐷𝑆 ↔ ∃𝑦𝑅 𝐵 = 𝐷)    &   (𝐴 = 𝐶 → (𝜑𝜓))    &   (𝐵 = 𝐷 → (𝜓𝜒))    &   ((𝑥𝑅𝑦𝑅) → 𝜑)       ((𝐶𝑆𝐷𝑆) → 𝜒)
 
Theorem3gencl 2646* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
(𝐷𝑆 ↔ ∃𝑥𝑅 𝐴 = 𝐷)    &   (𝐹𝑆 ↔ ∃𝑦𝑅 𝐵 = 𝐹)    &   (𝐺𝑆 ↔ ∃𝑧𝑅 𝐶 = 𝐺)    &   (𝐴 = 𝐷 → (𝜑𝜓))    &   (𝐵 = 𝐹 → (𝜓𝜒))    &   (𝐶 = 𝐺 → (𝜒𝜃))    &   ((𝑥𝑅𝑦𝑅𝑧𝑅) → 𝜑)       ((𝐷𝑆𝐹𝑆𝐺𝑆) → 𝜃)
 
Theoremcgsexg 2647* Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.)
(𝑥 = 𝐴𝜒)    &   (𝜒 → (𝜑𝜓))       (𝐴𝑉 → (∃𝑥(𝜒𝜑) ↔ 𝜓))
 
Theoremcgsex2g 2648* Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.)
((𝑥 = 𝐴𝑦 = 𝐵) → 𝜒)    &   (𝜒 → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊) → (∃𝑥𝑦(𝜒𝜑) ↔ 𝜓))
 
Theoremcgsex4g 2649* An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.)
(((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) → 𝜒)    &   (𝜒 → (𝜑𝜓))       (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑅𝐷𝑆)) → (∃𝑥𝑦𝑧𝑤(𝜒𝜑) ↔ 𝜓))
 
Theoremceqsex 2650* 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 = 𝐴𝜑) ↔ 𝜓)
 
Theoremceqsexv 2651* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
 
Theoremceqsex2 2652* 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 = 𝐴𝑦 = 𝐵𝜑) ↔ 𝜒)
 
Theoremceqsex2v 2653* 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 = 𝐴𝑦 = 𝐵𝜑) ↔ 𝜒)
 
Theoremceqsex3v 2654* 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 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑) ↔ 𝜃)
 
Theoremceqsex4v 2655* 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 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷) ∧ 𝜑) ↔ 𝜏)
 
Theoremceqsex6v 2656* 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 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ (𝑤 = 𝐷𝑣 = 𝐸𝑢 = 𝐹) ∧ 𝜑) ↔ 𝜁)
 
Theoremceqsex8v 2657* 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 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻)) ∧ 𝜑) ↔ 𝜌)
 
Theoremgencbvex 2658* 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))    &   (𝜃 ↔ ∃𝑥(𝜒𝐴 = 𝑦))       (∃𝑥(𝜒𝜑) ↔ ∃𝑦(𝜃𝜓))
 
Theoremgencbvex2 2659* Restatement of gencbvex 2658 with weaker hypotheses. (Contributed by Jeff Hankins, 6-Dec-2006.)
𝐴 ∈ V    &   (𝐴 = 𝑦 → (𝜑𝜓))    &   (𝐴 = 𝑦 → (𝜒𝜃))    &   (𝜃 → ∃𝑥(𝜒𝐴 = 𝑦))       (∃𝑥(𝜒𝜑) ↔ ∃𝑦(𝜃𝜓))
 
Theoremgencbval 2660* 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))    &   (𝜃 ↔ ∃𝑥(𝜒𝐴 = 𝑦))       (∀𝑥(𝜒𝜑) ↔ ∀𝑦(𝜃𝜓))
 
Theoremsbhypf 2661* 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 2662 Closed theorem form of vtoclgf 2670. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.)
(((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴𝑉) → 𝜓)
 
Theoremvtocldf 2663 Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))    &   (𝜑𝜓)    &   𝑥𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜒)       (𝜑𝜒)
 
Theoremvtocld 2664* Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))    &   (𝜑𝜓)       (𝜑𝜒)
 
Theoremvtoclf 2665* Implicit substitution of a class for a setvar variable. This is a generalization of chvar 1684. (Contributed by NM, 30-Aug-1993.)
𝑥𝜓    &   𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       𝜓
 
Theoremvtocl 2666* Implicit substitution of a class for a setvar variable. (Contributed by NM, 30-Aug-1993.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       𝜓
 
Theoremvtocl2 2667* 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 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))    &   𝜑       𝜓
 
Theoremvtocl3 2668* 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 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))    &   𝜑       𝜓
 
Theoremvtoclb 2669* 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 2670 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       (𝐴𝑉𝜓)
 
Theoremvtoclg 2671* Implicit substitution of a class expression for a setvar variable. (Contributed by NM, 17-Apr-1995.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       (𝐴𝑉𝜓)
 
Theoremvtoclbg 2672* Implicit substitution of a class for a setvar variable. (Contributed by NM, 29-Apr-1994.)
(𝑥 = 𝐴 → (𝜑𝜒))    &   (𝑥 = 𝐴 → (𝜓𝜃))    &   (𝜑𝜓)       (𝐴𝑉 → (𝜒𝜃))
 
Theoremvtocl2gf 2673 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       ((𝐴𝑉𝐵𝑊) → 𝜒)
 
Theoremvtocl3gf 2674 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       ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝜃)
 
Theoremvtocl2g 2675* Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 25-Apr-1995.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   𝜑       ((𝐴𝑉𝐵𝑊) → 𝜒)
 
Theoremvtoclgaf 2676* 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))    &   (𝑥𝐵𝜑)       (𝐴𝐵𝜓)
 
Theoremvtoclga 2677* Implicit substitution of a class for a setvar variable. (Contributed by NM, 20-Aug-1995.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥𝐵𝜑)       (𝐴𝐵𝜓)
 
Theoremvtocl2gaf 2678* 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))    &   ((𝑥𝐶𝑦𝐷) → 𝜑)       ((𝐴𝐶𝐵𝐷) → 𝜒)
 
Theoremvtocl2ga 2679* Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 20-Aug-1995.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   ((𝑥𝐶𝑦𝐷) → 𝜑)       ((𝐴𝐶𝐵𝐷) → 𝜒)
 
Theoremvtocl3gaf 2680* 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))    &   ((𝑥𝑅𝑦𝑆𝑧𝑇) → 𝜑)       ((𝐴𝑅𝐵𝑆𝐶𝑇) → 𝜃)
 
Theoremvtocl3ga 2681* Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   (𝑧 = 𝐶 → (𝜒𝜃))    &   ((𝑥𝐷𝑦𝑅𝑧𝑆) → 𝜑)       ((𝐴𝐷𝐵𝑅𝐶𝑆) → 𝜃)
 
Theoremvtocleg 2682* Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Jan-2004.)
(𝑥 = 𝐴𝜑)       (𝐴𝑉𝜑)
 
Theoremvtoclegft 2683* Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 2684.) (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)
((𝐴𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴𝜑)) → 𝜑)
 
Theoremvtoclef 2684* Implicit substitution of a class for a setvar variable. (Contributed by NM, 18-Aug-1993.)
𝑥𝜑    &   𝐴 ∈ V    &   (𝑥 = 𝐴𝜑)       𝜑
 
Theoremvtocle 2685* Implicit substitution of a class for a setvar variable. (Contributed by NM, 9-Sep-1993.)
𝐴 ∈ V    &   (𝑥 = 𝐴𝜑)       𝜑
 
Theoremvtoclri 2686* Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Nov-1994.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   𝑥𝐵 𝜑       (𝐴𝐵𝜓)
 
Theoremspcimgft 2687 A closed version of spcimgf 2691. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑥𝜓    &   𝑥𝐴       (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (∀𝑥𝜑𝜓)))
 
Theoremspcgft 2688 A closed version of spcgf 2693. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 4-Jan-2017.)
𝑥𝜓    &   𝑥𝐴       (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (∀𝑥𝜑𝜓)))
 
Theoremspcimegft 2689 A closed version of spcimegf 2692. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑥𝜓    &   𝑥𝐴       (∀𝑥(𝑥 = 𝐴 → (𝜓𝜑)) → (𝐴𝐵 → (𝜓 → ∃𝑥𝜑)))
 
Theoremspcegft 2690 A closed version of spcegf 2694. (Contributed by Jim Kingdon, 22-Jun-2018.)
𝑥𝜓    &   𝑥𝐴       (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (𝜓 → ∃𝑥𝜑)))
 
Theoremspcimgf 2691 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))       (𝐴𝑉 → (∀𝑥𝜑𝜓))
 
Theoremspcimegf 2692 Existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜓𝜑))       (𝐴𝑉 → (𝜓 → ∃𝑥𝜑))
 
Theoremspcgf 2693 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))       (𝐴𝑉 → (∀𝑥𝜑𝜓))
 
Theoremspcegf 2694 Existential specialization, using implicit substitution. (Contributed by NM, 2-Feb-1997.)
𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (𝜓 → ∃𝑥𝜑))
 
Theoremspcimdv 2695* Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓𝜒))
 
Theoremspcdv 2696* Rule of specialization, using implicit substitution. Analogous to rspcdv 2717. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓𝜒))
 
Theoremspcimedv 2697* Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜒𝜓))       (𝜑 → (𝜒 → ∃𝑥𝜓))
 
Theoremspcgv 2698* Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥𝜑𝜓))
 
Theoremspcegv 2699* Existential specialization, using implicit substitution. (Contributed by NM, 14-Aug-1994.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (𝜓 → ∃𝑥𝜑))
 
Theoremspc2egv 2700* Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊) → (𝜓 → ∃𝑥𝑦𝜑))
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