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Theorem List for Metamath Proof Explorer - 3501-3600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremspcdv 3501* Rule of specialization, using implicit substitution. Analogous to rspcdv 3521. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓𝜒))
 
Theoremspcimedv 3502* Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜒𝜓))       (𝜑 → (𝜒 → ∃𝑥𝜓))
 
Theoremspcgv 3503* Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.) Avoid ax-10 2145, ax-11 2162. (Revised by Wolf Lammen, 25-Aug-2023.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥𝜑𝜓))
 
Theoremspcegv 3504* Existential specialization, using implicit substitution. (Contributed by NM, 14-Aug-1994.) Avoid ax-10 2145, ax-11 2162. (Revised by Wolf Lammen, 25-Aug-2023.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (𝜓 → ∃𝑥𝜑))
 
Theoremspcedv 3505* Existential specialization, using implicit substitution, deduction version. (Contributed by RP, 12-Aug-2020.) (Revised by AV, 16-Aug-2024.)
(𝜑𝑋𝑉)    &   (𝜑𝜒)    &   (𝑥 = 𝑋 → (𝜓𝜒))       (𝜑 → ∃𝑥𝜓)
 
Theoremspc2egv 3506* Existential specialization with two quantifiers, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊) → (𝜓 → ∃𝑥𝑦𝜑))
 
Theoremspc2gv 3507* Specialization with two quantifiers, using implicit substitution. (Contributed by NM, 27-Apr-2004.)
((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊) → (∀𝑥𝑦𝜑𝜓))
 
Theoremspc2ed 3508* Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by Thierry Arnoux, 23-Aug-2017.)
𝑥𝜒    &   𝑦𝜒    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))       ((𝜑 ∧ (𝐴𝑉𝐵𝑊)) → (𝜒 → ∃𝑥𝑦𝜓))
 
Theoremspc2d 3509* Specialization with 2 quantifiers, using implicit substitution. (Contributed by Thierry Arnoux, 23-Aug-2017.)
𝑥𝜒    &   𝑦𝜒    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))       ((𝜑 ∧ (𝐴𝑉𝐵𝑊)) → (∀𝑥𝑦𝜓𝜒))
 
Theoremspc3egv 3510* Existential specialization with three quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.) Avoid ax-10 2145 and ax-11 2162. (Revised by Gino Giotto, 20-Aug-2023.) (Proof shortened by Wolf Lammen, 25-Aug-2023.)
((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝜓 → ∃𝑥𝑦𝑧𝜑))
 
Theoremspc3gv 3511* Specialization with three quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)
((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∀𝑥𝑦𝑧𝜑𝜓))
 
Theoremspcv 3512* Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥𝜑𝜓)
 
Theoremspcev 3513* Existential specialization, using implicit substitution. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝜓 → ∃𝑥𝜑)
 
Theoremspc2ev 3514* Existential specialization, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
𝐴 ∈ V    &   𝐵 ∈ V    &   ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       (𝜓 → ∃𝑥𝑦𝜑)
 
Theoremrspct 3515* A closed version of rspc 3517. (Contributed by Andrew Salmon, 6-Jun-2011.)
𝑥𝜓       (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓)))
 
Theoremrspcdf 3516* Restricted specialization, using implicit substitution. (Contributed by Emmett Weisz, 16-Jan-2020.)
𝑥𝜑    &   𝑥𝜒    &   (𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐵 𝜓𝜒))
 
Theoremrspc 3517* Restricted specialization, using implicit substitution. (Contributed by NM, 19-Apr-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))
 
Theoremrspce 3518* Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) (Revised by Mario Carneiro, 11-Oct-2016.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)
 
Theoremrspcimdv 3519* Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐵 𝜓𝜒))
 
Theoremrspcimedv 3520* Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜒𝜓))       (𝜑 → (𝜒 → ∃𝑥𝐵 𝜓))
 
Theoremrspcdv 3521* Restricted specialization, using implicit substitution. (Contributed by NM, 17-Feb-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐵 𝜓𝜒))
 
Theoremrspcedv 3522* Restricted existential specialization, using implicit substitution. (Contributed by FL, 17-Apr-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (𝜒 → ∃𝑥𝐵 𝜓))
 
Theoremrspcebdv 3523* Restricted existential specialization, using implicit substitution in both directions. (Contributed by AV, 8-Jan-2022.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))    &   ((𝜑𝜓) → 𝑥 = 𝐴)       (𝜑 → (∃𝑥𝐵 𝜓𝜒))
 
Theoremrspcv 3524* Restricted specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) Drop ax-10 2145, ax-11 2162, ax-12 2179. (Revised by SN, 12-Dec-2023.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))
 
TheoremrspcvOLD 3525* Obsolete version of rspcv 3524 as of 12-Dec-2023. Restricted specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))
 
Theoremrspccv 3526* Restricted specialization, using implicit substitution. (Contributed by NM, 2-Feb-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥𝐵 𝜑 → (𝐴𝐵𝜓))
 
Theoremrspcva 3527* Restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-2005.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵 ∧ ∀𝑥𝐵 𝜑) → 𝜓)
 
Theoremrspccva 3528* Restricted specialization, using implicit substitution. (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((∀𝑥𝐵 𝜑𝐴𝐵) → 𝜓)
 
Theoremrspcev 3529* Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) Drop ax-10 2145, ax-11 2162, ax-12 2179. (Revised by SN, 12-Dec-2023.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)
 
TheoremrspcevOLD 3530* Obsolete version of rspce 3518 as of 12-Dec-2023. Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)
 
Theoremrspcdva 3531* Restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 21-Jun-2020.)
(𝑥 = 𝐶 → (𝜓𝜒))    &   (𝜑 → ∀𝑥𝐴 𝜓)    &   (𝜑𝐶𝐴)       (𝜑𝜒)
 
Theoremrspcedvd 3532* Restricted existential specialization, using implicit substitution. Variant of rspcedv 3522. (Contributed by AV, 27-Nov-2019.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))    &   (𝜑𝜒)       (𝜑 → ∃𝑥𝐵 𝜓)
 
Theoremrspcime 3533* Prove a restricted existential. (Contributed by Rohan Ridenour, 3-Aug-2023.)
((𝜑𝑥 = 𝐴) → 𝜓)    &   (𝜑𝐴𝐵)       (𝜑 → ∃𝑥𝐵 𝜓)
 
Theoremrspceaimv 3534* Restricted existential specialization of a universally quantified implication. (Contributed by BJ, 24-Aug-2022.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵 ∧ ∀𝑦𝐶 (𝜓𝜒)) → ∃𝑥𝐵𝑦𝐶 (𝜑𝜒))
 
Theoremrspcedeq1vd 3535* Restricted existential specialization, using implicit substitution. Variant of rspcedvd 3532 for equations, in which the left hand side depends on the quantified variable. (Contributed by AV, 24-Dec-2019.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → 𝐶 = 𝐷)       (𝜑 → ∃𝑥𝐵 𝐶 = 𝐷)
 
Theoremrspcedeq2vd 3536* Restricted existential specialization, using implicit substitution. Variant of rspcedvd 3532 for equations, in which the right hand side depends on the quantified variable. (Contributed by AV, 24-Dec-2019.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → 𝐶 = 𝐷)       (𝜑 → ∃𝑥𝐵 𝐶 = 𝐷)
 
Theoremrspc2 3537* Restricted specialization with two quantifiers, using implicit substitution. (Contributed by NM, 9-Nov-2012.)
𝑥𝜒    &   𝑦𝜓    &   (𝑥 = 𝐴 → (𝜑𝜒))    &   (𝑦 = 𝐵 → (𝜒𝜓))       ((𝐴𝐶𝐵𝐷) → (∀𝑥𝐶𝑦𝐷 𝜑𝜓))
 
Theoremrspc2gv 3538* Restricted specialization with two quantifiers, using implicit substitution. (Contributed by BJ, 2-Dec-2021.)
((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊) → (∀𝑥𝑉𝑦𝑊 𝜑𝜓))
 
Theoremrspc2v 3539* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-1999.)
(𝑥 = 𝐴 → (𝜑𝜒))    &   (𝑦 = 𝐵 → (𝜒𝜓))       ((𝐴𝐶𝐵𝐷) → (∀𝑥𝐶𝑦𝐷 𝜑𝜓))
 
Theoremrspc2va 3540* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 18-Jun-2014.)
(𝑥 = 𝐴 → (𝜑𝜒))    &   (𝑦 = 𝐵 → (𝜒𝜓))       (((𝐴𝐶𝐵𝐷) ∧ ∀𝑥𝐶𝑦𝐷 𝜑) → 𝜓)
 
Theoremrspc2ev 3541* 2-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 16-Oct-1999.)
(𝑥 = 𝐴 → (𝜑𝜒))    &   (𝑦 = 𝐵 → (𝜒𝜓))       ((𝐴𝐶𝐵𝐷𝜓) → ∃𝑥𝐶𝑦𝐷 𝜑)
 
Theoremrspc3v 3542* 3-variable restricted specialization, using implicit substitution. (Contributed by NM, 10-May-2005.)
(𝑥 = 𝐴 → (𝜑𝜒))    &   (𝑦 = 𝐵 → (𝜒𝜃))    &   (𝑧 = 𝐶 → (𝜃𝜓))       ((𝐴𝑅𝐵𝑆𝐶𝑇) → (∀𝑥𝑅𝑦𝑆𝑧𝑇 𝜑𝜓))
 
Theoremrspc3ev 3543* 3-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 25-Jul-2012.)
(𝑥 = 𝐴 → (𝜑𝜒))    &   (𝑦 = 𝐵 → (𝜒𝜃))    &   (𝑧 = 𝐶 → (𝜃𝜓))       (((𝐴𝑅𝐵𝑆𝐶𝑇) ∧ 𝜓) → ∃𝑥𝑅𝑦𝑆𝑧𝑇 𝜑)
 
Theoremrspceeqv 3544* Restricted existential specialization in an equality, using implicit substitution. (Contributed by BJ, 2-Sep-2022.)
(𝑥 = 𝐴𝐶 = 𝐷)       ((𝐴𝐵𝐸 = 𝐷) → ∃𝑥𝐵 𝐸 = 𝐶)
 
Theoremralxpxfr2d 3545* Transfer a universal quantifier between one variable with pair-like semantics and two. (Contributed by Stefan O'Rear, 27-Feb-2015.)
𝐴 ∈ V    &   (𝜑 → (𝑥𝐵 ↔ ∃𝑦𝐶𝑧𝐷 𝑥 = 𝐴))    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐵 𝜓 ↔ ∀𝑦𝐶𝑧𝐷 𝜒))
 
Theoremrexraleqim 3546* Statement following from existence and generalization with equality. (Contributed by AV, 9-Feb-2019.)
(𝑥 = 𝑧 → (𝜓𝜑))    &   (𝑧 = 𝑌 → (𝜑𝜃))       ((∃𝑧𝐴 𝜑 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑌)) → 𝜃)
 
Theoremeqvincg 3547* A variable introduction law for class equality, closed form. (Contributed by Thierry Arnoux, 2-Mar-2017.)
(𝐴𝑉 → (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵)))
 
Theoremeqvinc 3548* A variable introduction law for class equality. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof shortened by Thierry Arnoux, 23-Jan-2022.)
𝐴 ∈ V       (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵))
 
Theoremeqvincf 3549 A variable introduction law for class equality, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Sep-2003.)
𝑥𝐴    &   𝑥𝐵    &   𝐴 ∈ V       (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵))
 
Theoremalexeqg 3550* Two ways to express substitution of 𝐴 for 𝑥 in 𝜑. This is the analogue for classes of sbalex 2244. (Contributed by NM, 2-Mar-1995.) (Revised by BJ, 27-Apr-2019.)
(𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
 
Theoremceqex 3551* Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.) (Proof shortened by BJ, 1-May-2019.)
(𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
 
Theoremceqsexg 3552* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 11-Oct-2004.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
 
Theoremceqsexgv 3553* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996.) Drop ax-10 2145 and ax-12 2179. (Revised by Gino Giotto, 1-Dec-2023.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
 
TheoremceqsexgvOLD 3554* Obsolete version of ceqsexgv 3553 as of 1-Dec-2023. (Contributed by NM, 29-Dec-1996.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
 
Theoremceqsrexv 3555* Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 30-Apr-2004.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝐵 → (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
 
Theoremceqsrexbv 3556* Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by Mario Carneiro, 14-Mar-2014.)
(𝑥 = 𝐴 → (𝜑𝜓))       (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ (𝐴𝐵𝜓))
 
Theoremceqsrex2v 3557* Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 29-Oct-2005.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))       ((𝐴𝐶𝐵𝐷) → (∃𝑥𝐶𝑦𝐷 ((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝜑) ↔ 𝜒))
 
Theoremclel2g 3558* Alternate definition of membership when the member is a set. (Contributed by NM, 18-Aug-1993.) Strengthen from sethood hypothesis to sethood antecedent. (Revised by BJ, 12-Feb-2022.) Avoid ax-12 2179. (Revised by BJ, 1-Sep-2024.)
(𝐴𝑉 → (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐴𝑥𝐵)))
 
Theoremclel2gOLD 3559* Obsolete version of clel2g 3558 as of 1-Sep-2024. (Contributed by NM, 18-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉 → (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐴𝑥𝐵)))
 
Theoremclel2 3560* Alternate definition of membership when the member is a set. (Contributed by NM, 18-Aug-1993.)
𝐴 ∈ V       (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐴𝑥𝐵))
 
Theoremclel3g 3561* Alternate definition of membership in a set. (Contributed by NM, 13-Aug-2005.)
(𝐵𝑉 → (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐵𝐴𝑥)))
 
Theoremclel3 3562* Alternate definition of membership in a set. (Contributed by NM, 18-Aug-1993.)
𝐵 ∈ V       (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐵𝐴𝑥))
 
Theoremclel4g 3563* Alternate definition of membership in a set. (Contributed by NM, 18-Aug-1993.) Strengthen from sethood hypothesis to sethood antecedent and avoid ax-12 2179. (Revised by BJ, 1-Sep-2024.)
(𝐵𝑉 → (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐵𝐴𝑥)))
 
Theoremclel4 3564* Alternate definition of membership in a set. (Contributed by NM, 18-Aug-1993.)
𝐵 ∈ V       (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐵𝐴𝑥))
 
Theoremclel4OLD 3565* Obsolete version of clel4 3564 as of 1-Sep-2024. (Contributed by NM, 18-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐵 ∈ V       (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐵𝐴𝑥))
 
Theoremclel5 3566* Alternate definition of class membership: a class 𝑋 is an element of another class 𝐴 iff there is an element of 𝐴 equal to 𝑋. (Contributed by AV, 13-Nov-2020.) Remove use of ax-10 2145, ax-11 2162, and ax-12 2179. (Revised by Steven Nguyen, 19-May-2023.)
(𝑋𝐴 ↔ ∃𝑥𝐴 𝑋 = 𝑥)
 
Theorempm13.183 3567* Compare theorem *13.183 in [WhiteheadRussell] p. 178. Only 𝐴 is required to be a set. (Contributed by Andrew Salmon, 3-Jun-2011.) Avoid ax-13 2373. (Revised by Wolf Lammen, 29-Apr-2023.)
(𝐴𝑉 → (𝐴 = 𝐵 ↔ ∀𝑧(𝑧 = 𝐴𝑧 = 𝐵)))
 
Theoremrr19.3v 3568* Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. We don't need the nonempty class condition of r19.3rzv 4395 when there is an outer quantifier. (Contributed by NM, 25-Oct-2012.)
(∀𝑥𝐴𝑦𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑)
 
Theoremrr19.28v 3569* Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. We don't need the nonempty class condition of r19.28zv 4397 when there is an outer quantifier. (Contributed by NM, 29-Oct-2012.)
(∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓))
 
Theoremelab6g 3570* Membership in a class abstraction. Class version of sb6 2095. (Contributed by SN, 5-Oct-2024.)
(𝐴𝑉 → (𝐴 ∈ {𝑥𝜑} ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
 
Theoremelabd2 3571* Membership in a class abstraction, using implicit substitution. Deduction version of elab 3578. (Contributed by Gino Giotto, 12-Oct-2024.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (𝐴 ∈ {𝑥𝜓} ↔ 𝜒))
 
Theoremelabgt 3572* Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 3576.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) Reduce axiom usage. (Revised by Gino Giotto, 12-Oct-2024.)
((𝐴𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
 
TheoremelabgtOLD 3573* Obsolete version of elabgt 3572 as of 12-Oct-2024. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
 
Theoremelabgf 3574 Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Revised by Mario Carneiro, 12-Oct-2016.)
𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝐵 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
 
Theoremelabf 3575* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 12-Oct-2016.)
𝑥𝜓    &   𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴 ∈ {𝑥𝜑} ↔ 𝜓)
 
Theoremelabg 3576* Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.) Avoid ax-13 2373. (Revised by SN, 23-Nov-2022.) Avoid ax-10 2145, ax-11 2162, ax-12 2179. (Revised by SN, 5-Oct-2024.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
 
TheoremelabgOLD 3577* Obsolete version of elabg 3576 as of 5-Oct-2024. (Contributed by NM, 14-Apr-1995.) Remove dependency on ax-13 2373. (Revised by SN, 23-Nov-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
 
Theoremelab 3578* Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.) Avoid ax-10 2145, ax-11 2162, ax-12 2179. (Revised by SN, 5-Oct-2024.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴 ∈ {𝑥𝜑} ↔ 𝜓)
 
TheoremelabOLD 3579* Obsolete version of elab 3578 as of 5-Oct-2024. (Contributed by NM, 1-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴 ∈ {𝑥𝜑} ↔ 𝜓)
 
Theoremelab2g 3580* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   𝐵 = {𝑥𝜑}       (𝐴𝑉 → (𝐴𝐵𝜓))
 
Theoremelabd 3581* Explicit demonstration the class {𝑥𝜓} is not empty by the example 𝐴. (Contributed by RP, 12-Aug-2020.) (Revised by AV, 23-Mar-2024.)
(𝜑𝐴𝑉)    &   (𝜑𝜒)    &   (𝑥 = 𝐴 → (𝜓𝜒))       (𝜑𝐴 ∈ {𝑥𝜓})
 
Theoremelab2 3582* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝐵 = {𝑥𝜑}       (𝐴𝐵𝜓)
 
Theoremelab4g 3583* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   𝐵 = {𝑥𝜑}       (𝐴𝐵 ↔ (𝐴 ∈ V ∧ 𝜓))
 
Theoremelab3gf 3584 Membership in a class abstraction, with a weaker antecedent than elabgf 3574. (Contributed by NM, 6-Sep-2011.)
𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       ((𝜓𝐴𝐵) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
 
Theoremelab3g 3585* Membership in a class abstraction, with a weaker antecedent than elabg 3576. (Contributed by NM, 29-Aug-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝜓𝐴𝐵) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
 
Theoremelab3 3586* Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.) (Revised by AV, 16-Aug-2024.)
(𝜓𝐴𝑉)    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴 ∈ {𝑥𝜑} ↔ 𝜓)
 
Theoremelrabi 3587* Implication for the membership in a restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.) Remove disjoint variable condition on 𝐴, 𝑥 and avoid ax-10 2145, ax-11 2162, ax-12 2179. (Revised by SN, 5-Aug-2024.)
(𝐴 ∈ {𝑥𝑉𝜑} → 𝐴𝑉)
 
TheoremelrabiOLD 3588* Obsolete version of elrabi 3587 as of 5-Aug-2024. (Contributed by Alexander van der Vekens, 31-Dec-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴 ∈ {𝑥𝑉𝜑} → 𝐴𝑉)
 
Theoremelrabf 3589 Membership in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵𝜓))
 
Theoremrabtru 3590 Abstract builder using the constant wff . (Contributed by Thierry Arnoux, 4-May-2020.)
𝑥𝐴       {𝑥𝐴 ∣ ⊤} = 𝐴
 
Theoremrabeqc 3591* A restricted class abstraction equals the restricting class if its condition follows from the membership of the free setvar variable in the restricting class. (Contributed by AV, 20-Apr-2022.)
(𝑥𝐴𝜑)       {𝑥𝐴𝜑} = 𝐴
 
Theoremelrab3t 3592* Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 3594.) (Contributed by Thierry Arnoux, 31-Aug-2017.)
((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝜓))
 
Theoremelrab 3593* Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 21-May-1999.) Remove dependency on ax-13 2373. (Revised by Steven Nguyen, 23-Nov-2022.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵𝜓))
 
Theoremelrab3 3594* Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝐵 → (𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝜓))
 
Theoremelrabd 3595* Membership in a restricted class abstraction, using implicit substitution. Deduction version of elrab 3593. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝑥 = 𝐴 → (𝜓𝜒))    &   (𝜑𝐴𝐵)    &   (𝜑𝜒)       (𝜑𝐴 ∈ {𝑥𝐵𝜓})
 
Theoremelrab2 3596* Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   𝐶 = {𝑥𝐵𝜑}       (𝐴𝐶 ↔ (𝐴𝐵𝜓))
 
Theoremralab 3597* Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝑦 = 𝑥 → (𝜑𝜓))       (∀𝑥 ∈ {𝑦𝜑}𝜒 ↔ ∀𝑥(𝜓𝜒))
 
Theoremralrab 3598* Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝑦 = 𝑥 → (𝜑𝜓))       (∀𝑥 ∈ {𝑦𝐴𝜑}𝜒 ↔ ∀𝑥𝐴 (𝜓𝜒))
 
Theoremrexab 3599* Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.)
(𝑦 = 𝑥 → (𝜑𝜓))       (∃𝑥 ∈ {𝑦𝜑}𝜒 ↔ ∃𝑥(𝜓𝜒))
 
Theoremrexrab 3600* Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.)
(𝑦 = 𝑥 → (𝜑𝜓))       (∃𝑥 ∈ {𝑦𝐴𝜑}𝜒 ↔ ∃𝑥𝐴 (𝜓𝜒))
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