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Theorem List for Metamath Proof Explorer - 3201-3300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcbvralv 3201* 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 3202* 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 3203* 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))       (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)

Theoremcbvrmov 3204* Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝐴 𝜑 ↔ ∃*𝑦𝐴 𝜓)

Theoremcbvraldva2 3205* 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 3206* 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 3207* 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 3208* 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 3209* 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 3210* 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 3211* 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 3212* 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 3213* 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 3214* Implicit to explicit substitution that swaps variables in a quantified expression. (Contributed by NM, 5-Sep-2004.)
(𝑥 = 𝑦 → (𝜑𝜓))       ([𝑥 / 𝑦]∀𝑥𝑦 𝜑 ↔ ∀𝑦𝑥 𝜓)

Theoremrabbiia 3215 Equivalent wff's yield equal restricted class abstractions (inference rule). (Contributed by NM, 22-May-1999.)
(𝑥𝐴 → (𝜑𝜓))       {𝑥𝐴𝜑} = {𝑥𝐴𝜓}

Theoremrabbii 3216 Equivalent wff's correspond to equal restricted class abstractions. Inference form of rabbidv 3220. (Contributed by Peter Mazsa, 1-Nov-2019.)
(𝜑𝜓)       {𝑥𝐴𝜑} = {𝑥𝐴𝜓}

Theoremrabbidva2 3217* Equivalent wff's yield equal restricted class abstractions. (Contributed by Thierry Arnoux, 4-Feb-2017.)
(𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})

Theoremrabbia2 3218 Equivalent wff's yield equal restricted class abstractions. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒))       {𝑥𝐴𝜓} = {𝑥𝐵𝜒}

Theoremrabbidva 3219* Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 28-Nov-2003.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})

Theoremrabbidv 3220* Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 10-Feb-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})

Theoremrabeqf 3221 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 3222 Equality theorem for restricted class abstractions. Inference form of rabeqf 3221. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐴    &   𝑥𝐵    &   𝐴 = 𝐵       {𝑥𝐴𝜑} = {𝑥𝐵𝜑}

Theoremrabeq 3223* Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.)
(𝐴 = 𝐵 → {𝑥𝐴𝜑} = {𝑥𝐵𝜑})

Theoremrabeqi 3224* Equality theorem for restricted class abstractions. Inference form of rabeq 3223. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐴 = 𝐵       {𝑥𝐴𝜑} = {𝑥𝐵𝜑}

Theoremrabeqdv 3225* Equality of restricted class abstractions. Deduction form of rabeq 3223. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜓})

Theoremrabeqbidv 3226* Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})

Theoremrabeqbidva 3227* Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})

Theoremrabeq2i 3228 Inference rule from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.)
𝐴 = {𝑥𝐵𝜑}       (𝑥𝐴 ↔ (𝑥𝐵𝜑))

Theoremcbvrab 3229 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 3230* 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 3231 Extend class notation to include the universal class symbol.
class V

Theoremvjust 3232 Soundness justification theorem for df-v 3233. (Contributed by Rodolfo Medina, 27-Apr-2010.)
{𝑥𝑥 = 𝑥} = {𝑦𝑦 = 𝑦}

Definitiondf-v 3233 Define the universal class. Definition 5.20 of [TakeutiZaring] p. 21. Also Definition 2.9 of [Quine] p. 19. The class V can be described as the "class of all sets"; vprc 4829 proves that V is not itself a set in ZFC. We will frequently use the expression 𝐴 ∈ V as a short way to say "𝐴 is a set", and isset 3238 proves that this expression has the same meaning as 𝑥𝑥 = 𝐴. The class V is called the "von Neumann universe", however, the letter "V" is not a tribute to the name of von Neumann. The letter "V" was used earlier by Peano in 1889 for the universe of sets, where the letter V is derived from the word "Verum". Peano's notation V was adopted by Whitehead and Russell in Principia Mathematica for the class of all sets in 1910. For a general discussion of the theory of classes, see mmset.html#class. (Contributed by NM, 26-May-1993.)
V = {𝑥𝑥 = 𝑥}

Theoremvex 3234 All setvar variables are sets (see isset 3238). Theorem 6.8 of [Quine] p. 43. (Contributed by NM, 26-May-1993.)
𝑥 ∈ V

Theoremeqvf 3235 The universe contains every set. (Contributed by BJ, 15-Jul-2021.)
𝑥𝐴       (𝐴 = V ↔ ∀𝑥 𝑥𝐴)

Theoremeqv 3236* The universe contains every set. (Contributed by NM, 11-Sep-2006.)
(𝐴 = V ↔ ∀𝑥 𝑥𝐴)

Theoremabv 3237 The class of sets verifying a property is the universal class if and only if that property is a tautology. (Contributed by BJ, 19-Mar-2021.)
({𝑥𝜑} = V ↔ ∀𝑥𝜑)

Theoremisset 3238* Two ways to say "𝐴 is a set": A class 𝐴 is a member of the universal class V (see df-v 3233) 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 6995. Note that a class 𝐴 which is not a set is called a proper class. In some theorems, such as uniexg 6997, 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 2647 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 3239 A version of isset 3238 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 3240* A way to say "𝐴 is a set" (inference rule). (Contributed by NM, 24-Jun-1993.)
𝐴 ∈ V       𝑥 𝑥 = 𝐴

Theoremissetri 3241* A way to say "𝐴 is a set" (inference rule). (Contributed by NM, 21-Jun-1993.)
𝑥 𝑥 = 𝐴       𝐴 ∈ V

Theoremeqvisset 3242 A class equal to a variable is a set. Note the absence of dv condition, contrary to isset 3238 and issetri 3241. (Contributed by BJ, 27-Apr-2019.)
(𝑥 = 𝐴𝐴 ∈ V)

Theoremelex 3243 If a class is a member of another class, it is a set. Theorem 6.12 of [Quine] p. 44. (Contributed by NM, 26-May-1993.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
(𝐴𝐵𝐴 ∈ V)

Theoremelexi 3244 If a class is a member of another class, it is a set. (Contributed by NM, 11-Jun-1994.)
𝐴𝐵       𝐴 ∈ V

Theoremelexd 3245 If a class is a member of another class, it is a set. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴𝑉)       (𝜑𝐴 ∈ V)

Theoremelisset 3246* An element of a class exists. (Contributed by NM, 1-May-1995.)
(𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)

Theoremelex2 3247* If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.)
(𝐴𝐵 → ∃𝑥 𝑥𝐵)

Theoremelex22 3248* If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.)
((𝐴𝐵𝐴𝐶) → ∃𝑥(𝑥𝐵𝑥𝐶))

Theoremprcnel 3249 A proper class doesn't belong to any class. (Contributed by Glauco Siliprandi, 17-Aug-2020.) (Proof shortened by AV, 14-Nov-2020.)
𝐴 ∈ V → ¬ 𝐴𝑉)

Theoremralv 3250 A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
(∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑)

Theoremrexv 3251 An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
(∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)

Theoremreuv 3252 A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.)
(∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑)

Theoremrmov 3253 A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥𝜑)

Theoremrabab 3254 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 3255* 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 3256* 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 3257* Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
(∃𝑥𝑦𝐴 (𝜑𝜓) ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥𝜓))

Theoremrexcom4b 3258* Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
𝐵 ∈ V       (∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵) ↔ ∃𝑦𝐴 𝜑)

Theoremceqsalt 3259* Closed theorem version of ceqsalg 3261. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))

Theoremceqsralt 3260* Restricted quantifier version of ceqsalt 3259. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))

Theoremceqsalg 3261* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. For an alternate proof, see ceqsalgALT 3262. (Contributed by NM, 29-Oct-2003.) (Proof shortened by BJ, 29-Sep-2019.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))

TheoremceqsalgALT 3262* Alternate proof of ceqsalg 3261, not using ceqsalt 3259. (Contributed by NM, 29-Oct-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by BJ, 29-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))

Theoremceqsal 3263* 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 3264* 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 3265* Restricted quantifier version of ceqsalv 3264. (Contributed by NM, 21-Jun-2013.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝐵 → (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))

Theoremgencl 3266* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
(𝜃 ↔ ∃𝑥(𝜒𝐴 = 𝐵))    &   (𝐴 = 𝐵 → (𝜑𝜓))    &   (𝜒𝜑)       (𝜃𝜓)

Theorem2gencl 3267* 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 3268* 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 3269* Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.)
(𝑥 = 𝐴𝜒)    &   (𝜒 → (𝜑𝜓))       (𝐴𝑉 → (∃𝑥(𝜒𝜑) ↔ 𝜓))

Theoremcgsex2g 3270* Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.)
((𝑥 = 𝐴𝑦 = 𝐵) → 𝜒)    &   (𝜒 → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊) → (∃𝑥𝑦(𝜒𝜑) ↔ 𝜓))

Theoremcgsex4g 3271* 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 3272* 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 3273* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)

Theoremceqsexv2d 3274* Elimination of an existential quantifier, using implicit substitution. (Contributed by Thierry Arnoux, 10-Sep-2016.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜓       𝑥𝜑

Theoremceqsex2 3275* 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 3276* 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 3277* 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 3278* 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 3279* 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 3280* 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 3281* 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 3282* Restatement of gencbvex 3281 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 3283* Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.)
𝐴 ∈ V    &   (𝐴 = 𝑦 → (𝜑𝜓))    &   (𝐴 = 𝑦 → (𝜒𝜃))    &   (𝜃 ↔ ∃𝑥(𝜒𝐴 = 𝑦))       (∀𝑥(𝜒𝜑) ↔ ∀𝑦(𝜃𝜓))

Theoremsbhypf 3284* Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3585. (Contributed by Raph Levien, 10-Apr-2004.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓))

Theoremvtoclgft 3285 Closed theorem form of vtoclgf 3295. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by JJ, 11-Aug-2021.)
(((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴𝑉) → 𝜓)

TheoremvtoclgftOLD 3286 Obsolete proof of vtoclgft 3285 as of 11-Aug-2021. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
(((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴𝑉) → 𝜓)

Theoremvtocldf 3287 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 3288* Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))    &   (𝜑𝜓)       (𝜑𝜒)

Theoremvtoclf 3289* Implicit substitution of a class for a setvar variable. This is a generalization of chvar 2298. (Contributed by NM, 30-Aug-1993.)
𝑥𝜓    &   𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       𝜓

Theoremvtocl 3290* Implicit substitution of a class for a setvar variable. See also vtoclALT 3291. (Contributed by NM, 30-Aug-1993.) Removed dependency on ax-10 2059. (Revised by BJ, 29-Nov-2020.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       𝜓

TheoremvtoclALT 3291* Alternate proof of vtocl 3290. Shorter but requires more axioms. (Contributed by NM, 30-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       𝜓

Theoremvtocl2 3292* 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 3293* 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 3294* 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 3295 Implicit substitution of a class for a setvar variable, with bound-variable hypotheses in place of disjoint variable restrictions. (Contributed by NM, 21-Sep-2003.) (Proof shortened by Mario Carneiro, 10-Oct-2016.)
𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       (𝐴𝑉𝜓)

Theoremvtoclg1f 3296* Version of vtoclgf 3295 with one non-freeness hypothesis replaced with a dv condition, thus avoiding dependency on ax-11 2074 and ax-13 2282. (Contributed by BJ, 1-May-2019.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       (𝐴𝑉𝜓)

Theoremvtoclg 3297* Implicit substitution of a class expression for a setvar variable. (Contributed by NM, 17-Apr-1995.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       (𝐴𝑉𝜓)

Theoremvtoclbg 3298* 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 3299 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 3300 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)

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