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Theorem List for Intuitionistic Logic Explorer - 2601-2700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremraleq 2601* Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
(𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜑))
 
Theoremrexeq 2602* Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.)
(𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑))
 
Theoremreueq1 2603* Equality theorem for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.)
(𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜑))
 
Theoremrmoeq1 2604* Equality theorem for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜑))
 
Theoremraleqi 2605* Equality inference for restricted universal qualifier. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐴 = 𝐵       (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜑)
 
Theoremrexeqi 2606* Equality inference for restricted existential qualifier. (Contributed by Mario Carneiro, 23-Apr-2015.)
𝐴 = 𝐵       (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑)
 
Theoremraleqdv 2607* Equality deduction for restricted universal quantifier. (Contributed by NM, 13-Nov-2005.)
(𝜑𝐴 = 𝐵)       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜓))
 
Theoremrexeqdv 2608* Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.)
(𝜑𝐴 = 𝐵)       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜓))
 
Theoremraleqbi1dv 2609* Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
(𝐴 = 𝐵 → (𝜑𝜓))       (𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜓))
 
Theoremrexeqbi1dv 2610* Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.)
(𝐴 = 𝐵 → (𝜑𝜓))       (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜓))
 
Theoremreueqd 2611* Equality deduction for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.)
(𝐴 = 𝐵 → (𝜑𝜓))       (𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜓))
 
Theoremrmoeqd 2612* Equality deduction for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(𝐴 = 𝐵 → (𝜑𝜓))       (𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜓))
 
Theoremraleqbidv 2613* Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))
 
Theoremrexeqbidv 2614* Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))
 
Theoremraleqbidva 2615* Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))
 
Theoremrexeqbidva 2616* Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))
 
Theoremmormo 2617 Unrestricted "at most one" implies restricted "at most one". (Contributed by NM, 16-Jun-2017.)
(∃*𝑥𝜑 → ∃*𝑥𝐴 𝜑)
 
Theoremreu5 2618 Restricted uniqueness in terms of "at most one." (Contributed by NM, 23-May-1999.) (Revised by NM, 16-Jun-2017.)
(∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∃*𝑥𝐴 𝜑))
 
Theoremreurex 2619 Restricted unique existence implies restricted existence. (Contributed by NM, 19-Aug-1999.)
(∃!𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜑)
 
Theoremreurmo 2620 Restricted existential uniqueness implies restricted "at most one." (Contributed by NM, 16-Jun-2017.)
(∃!𝑥𝐴 𝜑 → ∃*𝑥𝐴 𝜑)
 
Theoremrmo5 2621 Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.)
(∃*𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑))
 
Theoremnrexrmo 2622 Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.)
(¬ ∃𝑥𝐴 𝜑 → ∃*𝑥𝐴 𝜑)
 
Theoremcbvralf 2623 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))       (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)
 
Theoremcbvrexf 2624 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))       (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)
 
Theoremcbvral 2625* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)
 
Theoremcbvrex 2626* 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))       (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)
 
Theoremcbvreu 2627* Change the bound variable of a restricted unique existential quantifier using implicit substitution. (Contributed by Mario Carneiro, 15-Oct-2016.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
 
Theoremcbvrmo 2628* Change the bound variable of restricted "at most one" using implicit substitution. (Contributed by NM, 16-Jun-2017.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝐴 𝜑 ↔ ∃*𝑦𝐴 𝜓)
 
Theoremcbvralv 2629* Change the bound variable of a restricted universal quantifier using implicit substitution. (Contributed by NM, 28-Jan-1997.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)
 
Theoremcbvrexv 2630* Change the bound variable of a restricted existential quantifier using implicit substitution. (Contributed by NM, 2-Jun-1998.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)
 
Theoremcbvreuv 2631* Change the bound variable of a restricted unique existential quantifier using implicit substitution. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
 
Theoremcbvrmov 2632* Change the bound variable of a restricted at-most-one quantifier using implicit substitution. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝐴 𝜑 ↔ ∃*𝑦𝐴 𝜓)
 
Theoremcbvraldva2 2633* 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.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))    &   ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑦𝐵 𝜒))
 
Theoremcbvrexdva2 2634* 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.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))    &   ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐵 𝜒))
 
Theoremcbvraldva 2635* Rule used to change the bound variable in a restricted universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑦𝐴 𝜒))
 
Theoremcbvrexdva 2636* Rule used to change the bound variable in a restricted existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐴 𝜒))
 
Theoremcbvral2v 2637* Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by NM, 10-Aug-2004.)
(𝑥 = 𝑧 → (𝜑𝜒))    &   (𝑦 = 𝑤 → (𝜒𝜓))       (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑧𝐴𝑤𝐵 𝜓)
 
Theoremcbvrex2v 2638* Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by FL, 2-Jul-2012.)
(𝑥 = 𝑧 → (𝜑𝜒))    &   (𝑦 = 𝑤 → (𝜒𝜓))       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵 𝜓)
 
Theoremcbvral3v 2639* 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))       (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑤𝐴𝑣𝐵𝑢𝐶 𝜓)
 
Theoremcbvralsv 2640* Change bound variable by using a substitution. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.)
(∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 [𝑦 / 𝑥]𝜑)
 
Theoremcbvrexsv 2641* Change bound variable by using a substitution. (Contributed by NM, 2-Mar-2008.) (Revised by Andrew Salmon, 11-Jul-2011.)
(∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 [𝑦 / 𝑥]𝜑)
 
Theoremsbralie 2642* Implicit to explicit substitution that swaps variables in a quantified expression. (Contributed by NM, 5-Sep-2004.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝑦 𝜑 ↔ [𝑦 / 𝑥]∀𝑦𝑥 𝜓)
 
Theoremrabbiia 2643 Equivalent wff's yield equal restricted class abstractions (inference form). (Contributed by NM, 22-May-1999.)
(𝑥𝐴 → (𝜑𝜓))       {𝑥𝐴𝜑} = {𝑥𝐴𝜓}
 
Theoremrabbii 2644 Equivalent wff's correspond to equal restricted class abstractions. Inference form of rabbidv 2647. (Contributed by Peter Mazsa, 1-Nov-2019.)
(𝜑𝜓)       {𝑥𝐴𝜑} = {𝑥𝐴𝜓}
 
Theoremrabbidva2 2645* Equivalent wff's yield equal restricted class abstractions. (Contributed by Thierry Arnoux, 4-Feb-2017.)
(𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
 
Theoremrabbidva 2646* Equivalent wff's yield equal restricted class abstractions (deduction form). (Contributed by NM, 28-Nov-2003.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
 
Theoremrabbidv 2647* Equivalent wff's yield equal restricted class abstractions (deduction form). (Contributed by NM, 10-Feb-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
 
Theoremrabeqf 2648 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 2649 Equality theorem for restricted class abstractions. Inference form of rabeqf 2648. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐴    &   𝑥𝐵    &   𝐴 = 𝐵       {𝑥𝐴𝜑} = {𝑥𝐵𝜑}
 
Theoremrabeq 2650* Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.)
(𝐴 = 𝐵 → {𝑥𝐴𝜑} = {𝑥𝐵𝜑})
 
Theoremrabeqi 2651* Equality theorem for restricted class abstractions. Inference form of rabeq 2650. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐴 = 𝐵       {𝑥𝐴𝜑} = {𝑥𝐵𝜑}
 
Theoremrabeqdv 2652* Equality of restricted class abstractions. Deduction form of rabeq 2650. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜓})
 
Theoremrabeqbidv 2653* Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
 
Theoremrabeqbidva 2654* Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
 
Theoremrabeq2i 2655 Inference from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.)
𝐴 = {𝑥𝐵𝜑}       (𝑥𝐴 ↔ (𝑥𝐵𝜑))
 
Theoremcbvrab 2656 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 2657* 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 2658 Extend class notation to include the universal class symbol.
class V
 
Theoremvjust 2659 Soundness justification theorem for df-v 2660. (Contributed by Rodolfo Medina, 27-Apr-2010.)
{𝑥𝑥 = 𝑥} = {𝑦𝑦 = 𝑦}
 
Definitiondf-v 2660 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 2661 All setvar variables are sets (see isset 2664). Theorem 6.8 of [Quine] p. 43. (Contributed by NM, 5-Aug-1993.)
𝑥 ∈ V
 
Theoremelv 2662 Technical lemma used to shorten proofs. If a proposition is implied by 𝑥 ∈ V (which is true, see vex 2661), then it is true. (Contributed by Peter Mazsa, 13-Oct-2018.)
(𝑥 ∈ V → 𝜑)       𝜑
 
Theoremelvd 2663 Technical lemma used to shorten proofs. If a proposition is implied by 𝑥 ∈ V (which is true, see vex 2661) and another antecedent, then it is implied by the other antecedent. (Contributed by Peter Mazsa, 23-Oct-2018.)
((𝜑𝑥 ∈ V) → 𝜓)       (𝜑𝜓)
 
Theoremisset 2664* Two ways to say "𝐴 is a set": A class 𝐴 is a member of the universal class V (see df-v 2660) 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 4327. Note the when 𝐴 is not a set, it is called a proper class. In some theorems, such as uniexg 4329, 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 2111 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 2665 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 2666* A way to say "𝐴 is a set" (inference form). (Contributed by NM, 5-Aug-1993.)
𝐴 ∈ V       𝑥 𝑥 = 𝐴
 
Theoremissetri 2667* A way to say "𝐴 is a set" (inference form). (Contributed by NM, 5-Aug-1993.)
𝑥 𝑥 = 𝐴       𝐴 ∈ V
 
Theoremeqvisset 2668 A class equal to a variable is a set. Note the absence of disjoint variable condition, contrary to isset 2664 and issetri 2667. (Contributed by BJ, 27-Apr-2019.)
(𝑥 = 𝐴𝐴 ∈ V)
 
Theoremelex 2669 If a class is a member of another class, then 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 2670 If a class is a member of another class, it is a set. (Contributed by NM, 11-Jun-1994.)
𝐴𝐵       𝐴 ∈ V
 
Theoremelexd 2671 If a class is a member of another class, it is a set. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴𝑉)       (𝜑𝐴 ∈ V)
 
Theoremelisset 2672* An element of a class exists. (Contributed by NM, 1-May-1995.)
(𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
 
Theoremelex22 2673* If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.)
((𝐴𝐵𝐴𝐶) → ∃𝑥(𝑥𝐵𝑥𝐶))
 
Theoremelex2 2674* If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.)
(𝐴𝐵 → ∃𝑥 𝑥𝐵)
 
Theoremralv 2675 A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
(∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑)
 
Theoremrexv 2676 An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
(∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)
 
Theoremreuv 2677 A unique existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.)
(∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑)
 
Theoremrmov 2678 An at-most-one quantifier restricted to the universe is unrestricted. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥𝜑)
 
Theoremrabab 2679 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 2680* 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 2681* 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 2682* Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
(∃𝑥𝑦𝐴 (𝜑𝜓) ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥𝜓))
 
Theoremrexcom4b 2683* Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
𝐵 ∈ V       (∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵) ↔ ∃𝑦𝐴 𝜑)
 
Theoremceqsalt 2684* Closed theorem version of ceqsalg 2686. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
 
Theoremceqsralt 2685* Restricted quantifier version of ceqsalt 2684. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
 
Theoremceqsalg 2686* 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 2687* 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 2688* 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 2689* Restricted quantifier version of ceqsalv 2688. (Contributed by NM, 21-Jun-2013.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝐵 → (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
 
Theoremgencl 2690* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
(𝜃 ↔ ∃𝑥(𝜒𝐴 = 𝐵))    &   (𝐴 = 𝐵 → (𝜑𝜓))    &   (𝜒𝜑)       (𝜃𝜓)
 
Theorem2gencl 2691* 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 2692* 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 2693* Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.)
(𝑥 = 𝐴𝜒)    &   (𝜒 → (𝜑𝜓))       (𝐴𝑉 → (∃𝑥(𝜒𝜑) ↔ 𝜓))
 
Theoremcgsex2g 2694* Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.)
((𝑥 = 𝐴𝑦 = 𝐵) → 𝜒)    &   (𝜒 → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊) → (∃𝑥𝑦(𝜒𝜑) ↔ 𝜓))
 
Theoremcgsex4g 2695* 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 2696* 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 2697* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
 
Theoremceqsex2 2698* 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 2699* 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 2700* 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)
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