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Theorem List for Metamath Proof Explorer - 3401-3500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrexeqbidv 3401* Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.) Remove usage of ax-10 2139, ax-11 2154, and ax-12 2170 and reduce distinct variable conditions. (Revised by Steven Nguyen, 30-Apr-2023.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))

Theoremraleqbi1dv 3402* Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.) (Proof shortened by Steven Nguyen, 5-May-2023.)
(𝐴 = 𝐵 → (𝜑𝜓))       (𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜓))

Theoremrexeqbi1dv 3403* Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.) (Proof shortened by Steven Nguyen, 5-May-2023.)
(𝐴 = 𝐵 → (𝜑𝜓))       (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜓))

Theoremraleq 3404* Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.) Remove usage of ax-10 2139, ax-11 2154, and ax-12 2170. (Revised by Steven Nguyen, 30-Apr-2023.)
(𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜑))

Theoremrexeq 3405* Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.) Remove usage of ax-10 2139, ax-11 2154, and ax-12 2170. (Revised by Steven Nguyen, 30-Apr-2023.)
(𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑))

Theoremreueq1 3406* Equality theorem for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.) Remove usage of ax-10 2139, ax-11 2154, and ax-12 2170. (Revised by Steven Nguyen, 30-Apr-2023.)
(𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜑))

Theoremrmoeq1 3407* Equality theorem for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.) Remove usage of ax-10 2139, ax-11 2154, and ax-12 2170. (Revised by Steven Nguyen, 30-Apr-2023.)
(𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜑))

TheoremraleqOLD 3408* Obsolete version of raleq 3404 as of 5-May-2023. Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜑))

TheoremrexeqOLD 3409* Obsolete version of rexeq 3405 as of 5-May-2023. Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑))

Theoremreueq1OLD 3410* Obsolete version of reueq1 3406 as of 5-May-2023. Equality theorem for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜑))

Theoremrmoeq1OLD 3411* Obsolete version of rmoeq1 3407 as of 5-May-2023. Equality theorem for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜑))

Theoremraleqi 3412* Equality inference for restricted universal quantifier. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐴 = 𝐵       (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜑)

Theoremrexeqi 3413* Equality inference for restricted existential quantifier. (Contributed by Mario Carneiro, 23-Apr-2015.)
𝐴 = 𝐵       (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑)

Theoremraleqdv 3414* Equality deduction for restricted universal quantifier. (Contributed by NM, 13-Nov-2005.)
(𝜑𝐴 = 𝐵)       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜓))

Theoremrexeqdv 3415* Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.)
(𝜑𝐴 = 𝐵)       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜓))

Theoremraleqbi1dvOLD 3416* Obsolete version of raleqbi1dv 3402 as of 5-May-2023. Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴 = 𝐵 → (𝜑𝜓))       (𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜓))

Theoremrexeqbi1dvOLD 3417* Obsolete version of rexeqbi1dv 3403 as of 5-May-2023. Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴 = 𝐵 → (𝜑𝜓))       (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜓))

Theoremreueqd 3418* Equality deduction for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.)
(𝐴 = 𝐵 → (𝜑𝜓))       (𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜓))

Theoremrmoeqd 3419* 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))

Theoremraleqbid 3420 Equality deduction for restricted universal quantifier. (Contributed by Thierry Arnoux, 8-Mar-2017.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐵    &   (𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))

Theoremrexeqbid 3421 Equality deduction for restricted existential quantifier. (Contributed by Thierry Arnoux, 8-Mar-2017.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐵    &   (𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))

TheoremraleqbidvOLD 3422* Obsolete version of raleqbidv 3400 as of 30-Apr-2023. Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))

TheoremrexeqbidvOLD 3423* Obsolete version of rexeqbidv 3401 as of 30-Apr-2023. Equality deduction for restricted existential quantifier. (Contributed by NM, 6-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))

Theoremraleqbidva 3424* Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))

Theoremrexeqbidva 3425* Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))

Theoremraleleq 3426* All elements of a class are elements of a class equal to this class. (Contributed by AV, 30-Oct-2020.)
(𝐴 = 𝐵 → ∀𝑥𝐴 𝑥𝐵)

TheoremraleleqALT 3427* Alternate proof of raleleq 3426 using ralel 3147, being longer and using more axioms. (Contributed by AV, 30-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴 = 𝐵 → ∀𝑥𝐴 𝑥𝐵)

Theoremmormo 3428 Unrestricted "at most one" implies restricted "at most one". (Contributed by NM, 16-Jun-2017.)
(∃*𝑥𝜑 → ∃*𝑥𝐴 𝜑)

Theoremreu5 3429 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 3430 Restricted unique existence implies restricted existence. (Contributed by NM, 19-Aug-1999.)
(∃!𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜑)

Theorem2reu2rex 3431 Double restricted existential uniqueness, analogous to 2eu2ex 2722. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
(∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 → ∃𝑥𝐴𝑦𝐵 𝜑)

Theoremreurmo 3432 Restricted existential uniqueness implies restricted "at most one." (Contributed by NM, 16-Jun-2017.)
(∃!𝑥𝐴 𝜑 → ∃*𝑥𝐴 𝜑)

Theoremrmo5 3433 Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.)
(∃*𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑))

Theoremnrexrmo 3434 Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.)
(¬ ∃𝑥𝐴 𝜑 → ∃*𝑥𝐴 𝜑)

Theoremreueubd 3435* Restricted existential uniqueness is equivalent to existential uniqueness if the unique element is in the restricting class. (Contributed by AV, 4-Jan-2021.)
((𝜑𝜓) → 𝑥𝑉)       (𝜑 → (∃!𝑥𝑉 𝜓 ↔ ∃!𝑥𝜓))

Theoremcbvralfw 3436* Version of cbvralf 3438 with a disjoint variable condition, which does not require ax-13 2384. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)

Theoremcbvrexfw 3437* Version of cbvrexf 3439 with a disjoint variable condition, which does not require ax-13 2384. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)

Theoremcbvralf 3438 Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2384. Use the weaker cbvralfw 3436 when possible. (Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro, 9-Oct-2016.) (New usage is discouraged.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)

Theoremcbvrexf 3439 Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2384. Use the weaker cbvrexfw 3437 when possible. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 9-Oct-2016.) (New usage is discouraged.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)

Theoremcbvralw 3440* Version of cbvral 3444 with a disjoint variable condition, which does not require ax-13 2384. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)

Theoremcbvrexw 3441* Version of cbvrex 3445 with a disjoint variable condition, which does not require ax-13 2384. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)

Theoremcbvreuw 3442* Version of cbvreu 3446 with a disjoint variable condition, which does not require ax-13 2384. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)

Theoremcbvrmow 3443* Version of cbvrmo 3447 with a disjoint variable condition, which does not require ax-13 2384. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝐴 𝜑 ↔ ∃*𝑦𝐴 𝜓)

Theoremcbvral 3444* Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2384. Use the weaker cbvralw 3440 when possible. (Contributed by NM, 31-Jul-2003.) (New usage is discouraged.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)

Theoremcbvrex 3445* Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2384. Use the weaker cbvrexw 3441 when possible. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (New usage is discouraged.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)

Theoremcbvreu 3446* Change the bound variable of a restricted unique existential quantifier using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2384. Use the weaker cbvreuw 3442 when possible. (Contributed by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)

Theoremcbvrmo 3447* Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2384. Use the weaker cbvrmow 3443 when possible. (Contributed by NM, 16-Jun-2017.) (New usage is discouraged.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝐴 𝜑 ↔ ∃*𝑦𝐴 𝜓)

Theoremcbvralvw 3448* Version of cbvralv 3451 with a disjoint variable condition, which does not require ax-10 2139, ax-11 2154, ax-12 2170, ax-13 2384. (Contributed by Gino Giotto, 10-Jan-2024.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)

Theoremcbvrexvw 3449* Version of cbvrexv 3452 with a disjoint variable condition, which does not require ax-10 2139, ax-11 2154, ax-12 2170, ax-13 2384. (Contributed by Gino Giotto, 10-Jan-2024.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)

Theoremcbvreuvw 3450* Version of cbvreuv 3453 with a disjoint variable condition, which does not require ax-13 2384. (Contributed by Gino Giotto, 10-Jan-2024.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)

Theoremcbvralv 3451* Change the bound variable of a restricted universal quantifier using implicit substitution. See cbvralvw 3448 based on fewer axioms , but extra disjoint variables. Usage of this theorem is discouraged because it depends on ax-13 2384. Use the weaker cbvralvw 3448 when possible. (Contributed by NM, 28-Jan-1997.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)

Theoremcbvrexv 3452* Change the bound variable of a restricted existential quantifier using implicit substitution. See cbvrexvw 3449 based on fewer axioms , but extra disjoint variables. Usage of this theorem is discouraged because it depends on ax-13 2384. Use the weaker cbvrexvw 3449 when possible. (Contributed by NM, 2-Jun-1998.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)

Theoremcbvreuv 3453* Change the bound variable of a restricted unique existential quantifier using implicit substitution. See cbvreuvw 3450 for a version without ax-13 2384, but extra disjoint variables. Usage of this theorem is discouraged because it depends on ax-13 2384. Use the weaker cbvreuvw 3450 when possible. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)

Theoremcbvrmov 3454* Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2384. (Contributed by Alexander van der Vekens, 17-Jun-2017.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝐴 𝜑 ↔ ∃*𝑦𝐴 𝜓)

Theoremcbvraldva2 3455* 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 3456* 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.) (Proof shortened by Wolf Lammen, 12-Aug-2023.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))    &   ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐵 𝜒))

Theoremcbvrexdva2OLD 3457* Obsolete version of cbvrexdva 3459 as of 12-Aug-2023. (Contributed by David Moews, 1-May-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))    &   ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐵 𝜒))

Theoremcbvraldva 3458* 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 3459* 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))

Theoremcbvral2vw 3460* Version of cbvral2v 3463 with a disjoint variable condition, which does not require ax-13 2384. (Contributed by Gino Giotto, 10-Jan-2024.)
(𝑥 = 𝑧 → (𝜑𝜒))    &   (𝑦 = 𝑤 → (𝜒𝜓))       (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑧𝐴𝑤𝐵 𝜓)

Theoremcbvrex2vw 3461* Version of cbvrex2v 3464 with a disjoint variable condition, which does not require ax-13 2384. (Contributed by Gino Giotto, 10-Jan-2024.)
(𝑥 = 𝑧 → (𝜑𝜒))    &   (𝑦 = 𝑤 → (𝜒𝜓))       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵 𝜓)

Theoremcbvral3vw 3462* Version of cbvral3v 3465 with a disjoint variable condition, which does not require ax-13 2384. (Contributed by Gino Giotto, 10-Jan-2024.)
(𝑥 = 𝑤 → (𝜑𝜒))    &   (𝑦 = 𝑣 → (𝜒𝜃))    &   (𝑧 = 𝑢 → (𝜃𝜓))       (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑤𝐴𝑣𝐵𝑢𝐶 𝜓)

Theoremcbvral2v 3463* Change bound variables of double restricted universal quantification, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2384. Use the weaker cbvral2vw 3460 when possible. (Contributed by NM, 10-Aug-2004.) (New usage is discouraged.)
(𝑥 = 𝑧 → (𝜑𝜒))    &   (𝑦 = 𝑤 → (𝜒𝜓))       (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑧𝐴𝑤𝐵 𝜓)

Theoremcbvrex2v 3464* Change bound variables of double restricted universal quantification, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2384. Use the weaker cbvrex2vw 3461 when possible. (Contributed by FL, 2-Jul-2012.) (New usage is discouraged.)
(𝑥 = 𝑧 → (𝜑𝜒))    &   (𝑦 = 𝑤 → (𝜒𝜓))       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵 𝜓)

Theoremcbvral3v 3465* Change bound variables of triple restricted universal quantification, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2384. Use the weaker cbvral3vw 3462 when possible. (Contributed by NM, 10-May-2005.) (New usage is discouraged.)
(𝑥 = 𝑤 → (𝜑𝜒))    &   (𝑦 = 𝑣 → (𝜒𝜃))    &   (𝑧 = 𝑢 → (𝜃𝜓))       (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑤𝐴𝑣𝐵𝑢𝐶 𝜓)

Theoremcbvralsvw 3466* Version of cbvralsv 3468 with a disjoint variable condition, which does not require ax-13 2384. (Contributed by Gino Giotto, 10-Jan-2024.)
(∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 [𝑦 / 𝑥]𝜑)

Theoremcbvrexsvw 3467* Version of cbvrexsv 3469 with a disjoint variable condition, which does not require ax-13 2384. (Contributed by Gino Giotto, 10-Jan-2024.)
(∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 [𝑦 / 𝑥]𝜑)

Theoremcbvralsv 3468* Change bound variable by using a substitution. Usage of this theorem is discouraged because it depends on ax-13 2384. Use the weaker cbvralsvw 3466 when possible. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.) (New usage is discouraged.)
(∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 [𝑦 / 𝑥]𝜑)

Theoremcbvrexsv 3469* Change bound variable by using a substitution. Usage of this theorem is discouraged because it depends on ax-13 2384. Use the weaker cbvrexsvw 3467 when possible. (Contributed by NM, 2-Mar-2008.) (Revised by Andrew Salmon, 11-Jul-2011.) (New usage is discouraged.)
(∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 [𝑦 / 𝑥]𝜑)

Theoremsbralie 3470* 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 3471 Equivalent wff's yield equal restricted class abstractions (inference form). (Contributed by NM, 22-May-1999.)
(𝑥𝐴 → (𝜑𝜓))       {𝑥𝐴𝜑} = {𝑥𝐴𝜓}

Theoremrabbii 3472 Equivalent wff's correspond to equal restricted class abstractions. Inference form of rabbidv 3479. (Contributed by Peter Mazsa, 1-Nov-2019.)
(𝜑𝜓)       {𝑥𝐴𝜑} = {𝑥𝐴𝜓}

Theoremrabbida 3473 Equivalent wff's yield equal restricted class abstractions (deduction form). Version of rabbidva 3477 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})

Theoremrabbid 3474 Version of rabbidv 3479 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})

Theoremrabbidva2 3475* Equivalent wff's yield equal restricted class abstractions. (Contributed by Thierry Arnoux, 4-Feb-2017.)
(𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})

Theoremrabbia2 3476 Equivalent wff's yield equal restricted class abstractions. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒))       {𝑥𝐴𝜓} = {𝑥𝐵𝜒}

Theoremrabbidva 3477* Equivalent wff's yield equal restricted class abstractions (deduction form). (Contributed by NM, 28-Nov-2003.) (Proof shortened by SN, 3-Dec-2023.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})

TheoremrabbidvaOLD 3478* Obsolete proof of rabbidva 3477 as of 4-Dec-2023. (Contributed by NM, 28-Nov-2003.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})

Theoremrabbidv 3479* Equivalent wff's yield equal restricted class abstractions (deduction form). (Contributed by NM, 10-Feb-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})

Theoremrabeqf 3480 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})

Theoremrabeqi 3481 Equality theorem for restricted class abstractions. Inference form of rabeqf 3480. (Contributed by Glauco Siliprandi, 26-Jun-2021.) Avoid ax-10 2139 and ax-11 2154. (Revised by Gino Giotto, 20-Aug-2023.)
𝐴 = 𝐵       {𝑥𝐴𝜑} = {𝑥𝐵𝜑}

Theoremrabeq 3482* Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.) Avoid ax-10 2139, ax-11 2154, ax-12 2170. (Revised by Gino Giotto, 20-Aug-2023.)
(𝐴 = 𝐵 → {𝑥𝐴𝜑} = {𝑥𝐵𝜑})

Theoremrabeqdv 3483* Equality of restricted class abstractions. Deduction form of rabeq 3482. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜓})

Theoremrabeqbidv 3484* Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})

Theoremrabeqbidva 3485* Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})

Theoremrabeq2i 3486 Inference from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.)
𝐴 = {𝑥𝐵𝜑}       (𝑥𝐴 ↔ (𝑥𝐵𝜑))

Theoremrabswap 3487 Swap with a membership relation in a restricted class abstraction. (Contributed by NM, 4-Jul-2005.)
{𝑥𝐴𝑥𝐵} = {𝑥𝐵𝑥𝐴}

Theoremcbvrabw 3488* Version of cbvrab 3489 with a disjoint variable condition, which does not require ax-13 2384. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝐴𝜑} = {𝑦𝐴𝜓}

Theoremcbvrab 3489 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. Usage of this theorem is discouraged because it depends on ax-13 2384. Use the weaker cbvrabw 3488 when possible. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 9-Oct-2016.) (New usage is discouraged.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝐴𝜑} = {𝑦𝐴𝜓}

Theoremcbvrabv 3490* Rule to change the bound variable in a restricted class abstraction, using implicit substitution. (Contributed by NM, 26-May-1999.) Require 𝑥, 𝑦 be disjoint to avoid ax-11 2154 and ax-13 2384. (Revised by Steven Nguyen, 4-Dec-2022.)
(𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝐴𝜑} = {𝑦𝐴𝜓}

TheoremcbvrabvOLD 3491* Obsolete version of cbvrabv 3490 as of 14-Jun-2023. Rule to change the bound variable in a restricted class abstraction, using implicit substitution. (Contributed by NM, 26-May-1999.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝐴𝜑} = {𝑦𝐴𝜓}

Theoremrabrabi 3492* Abstract builder restricted to another restricted abstract builder with implicit substitution. (Contributed by AV, 2-Aug-2022.) Avoid ax-10 2139 and ax-11 2154. (Revised by Gino Giotto, 20-Aug-2023.)
(𝑥 = 𝑦 → (𝜒𝜑))       {𝑥 ∈ {𝑦𝐴𝜑} ∣ 𝜓} = {𝑥𝐴 ∣ (𝜒𝜓)}

2.1.6  The universal class

Syntaxcvv 3493 Extend class notation to include the universal class symbol.
class V

Theoremvjust 3494 Soundness justification theorem for df-v 3495. (Contributed by Rodolfo Medina, 27-Apr-2010.)
{𝑥𝑥 = 𝑥} = {𝑦𝑦 = 𝑦}

Definitiondf-v 3495 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 5210 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 3505 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 3505. (Contributed by NM, 26-May-1993.)
V = {𝑥𝑥 = 𝑥}

Theoremvex 3496 All setvar variables are sets (see isset 3505). Theorem 6.8 of [Quine] p. 43. (Contributed by NM, 26-May-1993.) Remove use of ax-12 2170. (Revised by SN, 28-Aug-2023.)
𝑥 ∈ V

TheoremvexOLD 3497 Obsolete version of vex 3496 as of 28-Aug-2023. All setvar variables are sets (see isset 3505). Theorem 6.8 of [Quine] p. 43. (Contributed by NM, 26-May-1993.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑥 ∈ V

Theoremelv 3498 If a proposition is implied by 𝑥 ∈ V (which is true, see vex 3496), then it is true. (Contributed by Peter Mazsa, 13-Oct-2018.)
(𝑥 ∈ V → 𝜑)       𝜑

Theoremelvd 3499 If a proposition is implied by 𝑥 ∈ V (which is true, see vex 3496) and another antecedent, then it is implied by that other antecedent. Deduction associated with elv 3498. (Contributed by Peter Mazsa, 23-Oct-2018.)
((𝜑𝑥 ∈ V) → 𝜓)       (𝜑𝜓)

Theoremel2v 3500 If a proposition is implied by 𝑥 ∈ V and 𝑦 ∈ V (which is true, see vex 3496), then it is true. (Contributed by Peter Mazsa, 13-Oct-2018.)
((𝑥 ∈ V ∧ 𝑦 ∈ V) → 𝜑)       𝜑

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