HomeHome Metamath Proof Explorer
Theorem List (p. 37 of 449)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-28623)
  Hilbert Space Explorer  Hilbert Space Explorer
(28624-30146)
  Users' Mathboxes  Users' Mathboxes
(30147-44804)
 

Theorem List for Metamath Proof Explorer - 3601-3700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremspc2ed 3601* 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 3602* 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 3603* Existential specialization with three quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.) Avoid ax-10 2136 and ax-11 2151. (Revised by Gino Giotto, 20-Aug-2023.) (Proof shortened by Wolf Lammen, 25-Aug-2023.)
((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝜓 → ∃𝑥𝑦𝑧𝜑))
 
Theoremspc3gv 3604* Specialization with three quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)
((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∀𝑥𝑦𝑧𝜑𝜓))
 
Theoremspcv 3605* Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥𝜑𝜓)
 
Theoremspcev 3606* 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 3607* Existential specialization, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
𝐴 ∈ V    &   𝐵 ∈ V    &   ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       (𝜓 → ∃𝑥𝑦𝜑)
 
Theoremrspct 3608* A closed version of rspc 3610. (Contributed by Andrew Salmon, 6-Jun-2011.)
𝑥𝜓       (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓)))
 
Theoremrspcdf 3609* Restricted specialization, using implicit substitution. (Contributed by Emmett Weisz, 16-Jan-2020.)
𝑥𝜑    &   𝑥𝜒    &   (𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐵 𝜓𝜒))
 
Theoremrspc 3610* 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 3611* 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 3612* Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐵 𝜓𝜒))
 
Theoremrspcimedv 3613* Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜒𝜓))       (𝜑 → (𝜒 → ∃𝑥𝐵 𝜓))
 
Theoremrspcdv 3614* 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 3615* 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 3616* 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 3617* Restricted specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) Drop ax-10 2136, ax-11 2151, ax-12 2167. (Revised by SN, 12-Dec-2023.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))
 
TheoremrspcvOLD 3618* Obsolete version of rspcv 3617 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 3619* Restricted specialization, using implicit substitution. (Contributed by NM, 2-Feb-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥𝐵 𝜑 → (𝐴𝐵𝜓))
 
Theoremrspcva 3620* Restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-2005.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵 ∧ ∀𝑥𝐵 𝜑) → 𝜓)
 
Theoremrspccva 3621* 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 3622* Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) Drop ax-10 2136, ax-11 2151, ax-12 2167. (Revised by SN, 12-Dec-2023.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)
 
TheoremrspcevOLD 3623* Obsolete version of rspce 3611 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 3624* Restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 21-Jun-2020.)
(𝑥 = 𝐶 → (𝜓𝜒))    &   (𝜑 → ∀𝑥𝐴 𝜓)    &   (𝜑𝐶𝐴)       (𝜑𝜒)
 
Theoremrspcedvd 3625* Restricted existential specialization, using implicit substitution. Variant of rspcedv 3615. (Contributed by AV, 27-Nov-2019.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))    &   (𝜑𝜒)       (𝜑 → ∃𝑥𝐵 𝜓)
 
Theoremrspcime 3626* Prove a restricted existential. (Contributed by Rohan Ridenour, 3-Aug-2023.)
((𝜑𝑥 = 𝐴) → 𝜓)    &   (𝜑𝐴𝐵)       (𝜑 → ∃𝑥𝐵 𝜓)
 
Theoremrspceaimv 3627* Restricted existential specialization of a universally quantified implication. (Contributed by BJ, 24-Aug-2022.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵 ∧ ∀𝑦𝐶 (𝜓𝜒)) → ∃𝑥𝐵𝑦𝐶 (𝜑𝜒))
 
Theoremrspcedeq1vd 3628* Restricted existential specialization, using implicit substitution. Variant of rspcedvd 3625 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 3629* Restricted existential specialization, using implicit substitution. Variant of rspcedvd 3625 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 3630* 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 3631* Restricted specialization with two quantifiers, using implicit substitution. (Contributed by BJ, 2-Dec-2021.)
((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊) → (∀𝑥𝑉𝑦𝑊 𝜑𝜓))
 
Theoremrspc2v 3632* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-1999.)
(𝑥 = 𝐴 → (𝜑𝜒))    &   (𝑦 = 𝐵 → (𝜒𝜓))       ((𝐴𝐶𝐵𝐷) → (∀𝑥𝐶𝑦𝐷 𝜑𝜓))
 
Theoremrspc2va 3633* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 18-Jun-2014.)
(𝑥 = 𝐴 → (𝜑𝜒))    &   (𝑦 = 𝐵 → (𝜒𝜓))       (((𝐴𝐶𝐵𝐷) ∧ ∀𝑥𝐶𝑦𝐷 𝜑) → 𝜓)
 
Theoremrspc2ev 3634* 2-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 16-Oct-1999.)
(𝑥 = 𝐴 → (𝜑𝜒))    &   (𝑦 = 𝐵 → (𝜒𝜓))       ((𝐴𝐶𝐵𝐷𝜓) → ∃𝑥𝐶𝑦𝐷 𝜑)
 
Theoremrspc3v 3635* 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 3636* 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 3637* Restricted existential specialization in an equality, using implicit substitution. (Contributed by BJ, 2-Sep-2022.)
(𝑥 = 𝐴𝐶 = 𝐷)       ((𝐴𝐵𝐸 = 𝐷) → ∃𝑥𝐵 𝐸 = 𝐶)
 
Theoremralxpxfr2d 3638* 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 3639* 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 3640* A variable introduction law for class equality, closed form. (Contributed by Thierry Arnoux, 2-Mar-2017.)
(𝐴𝑉 → (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵)))
 
Theoremeqvinc 3641* 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 3642 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 3643* Two ways to express substitution of 𝐴 for 𝑥 in 𝜑. This is the analogue for classes of sb56 2269. (Contributed by NM, 2-Mar-1995.) (Revised by BJ, 27-Apr-2019.)
(𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
 
Theoremceqex 3644* 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 3645* 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 3646* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996.) Drop ax-10 2136 and ax-12 2167. (Revised by Gino Giotto, 1-Dec-2023.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
 
TheoremceqsexgvOLD 3647* Obsolete version of ceqsexgv 3646 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 3648* Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 30-Apr-2004.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝐵 → (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
 
Theoremceqsrexbv 3649* Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by Mario Carneiro, 14-Mar-2014.)
(𝑥 = 𝐴 → (𝜑𝜓))       (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ (𝐴𝐵𝜓))
 
Theoremceqsrex2v 3650* 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 3651* An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.) (Revised by BJ, 12-Feb-2022.)
(𝐴𝑉 → (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐴𝑥𝐵)))
 
Theoremclel2 3652* An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
𝐴 ∈ V       (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐴𝑥𝐵))
 
Theoremclel3g 3653* An alternate definition of class membership when the class is a set. (Contributed by NM, 13-Aug-2005.)
(𝐵𝑉 → (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐵𝐴𝑥)))
 
Theoremclel3 3654* An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
𝐵 ∈ V       (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐵𝐴𝑥))
 
Theoremclel4 3655* An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
𝐵 ∈ V       (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐵𝐴𝑥))
 
Theoremclel5 3656* 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 2136, ax-11 2151, and ax-12 2167. (Revised by Steven Nguyen, 19-May-2023.)
(𝑋𝐴 ↔ ∃𝑥𝐴 𝑋 = 𝑥)
 
Theoremclel5OLD 3657* Obsolete version of clel5 3656 as of 19-May-2023. 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.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑋𝐴 ↔ ∃𝑥𝐴 𝑋 = 𝑥)
 
Theorempm13.183 3658* 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 2383. (Revised by Wolf Lammen, 29-Apr-2023.)
(𝐴𝑉 → (𝐴 = 𝐵 ↔ ∀𝑧(𝑧 = 𝐴𝑧 = 𝐵)))
 
Theorempm13.183OLD 3659* Obsolete version of pm13.183 3658 as of 29-Apr-2023. (Contributed by Andrew Salmon, 3-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉 → (𝐴 = 𝐵 ↔ ∀𝑧(𝑧 = 𝐴𝑧 = 𝐵)))
 
Theoremrr19.3v 3660* Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. We don't need the nonempty class condition of r19.3rzv 4442 when there is an outer quantifier. (Contributed by NM, 25-Oct-2012.)
(∀𝑥𝐴𝑦𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑)
 
Theoremrr19.28v 3661* Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. We don't need the nonempty class condition of r19.28zv 4444 when there is an outer quantifier. (Contributed by NM, 29-Oct-2012.)
(∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓))
 
Theoremelabgt 3662* Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 3665.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
((𝐴𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
 
Theoremelabgf 3663 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 3664* 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 3665* Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.) Remove dependency on ax-13 2383. (Revised by Steven Nguyen, 23-Nov-2022.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
 
Theoremelab 3666* Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴 ∈ {𝑥𝜑} ↔ 𝜓)
 
Theoremelabd 3667* Explicit demonstration the class {𝑥𝜓} is not empty by the example 𝑋. (Contributed by RP, 12-Aug-2020.)
(𝜑𝑋 ∈ V)    &   (𝜑𝜒)    &   (𝑥 = 𝑋 → (𝜓𝜒))       (𝜑 → ∃𝑥𝜓)
 
Theoremelab2g 3668* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   𝐵 = {𝑥𝜑}       (𝐴𝑉 → (𝐴𝐵𝜓))
 
Theoremelab2 3669* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝐵 = {𝑥𝜑}       (𝐴𝐵𝜓)
 
Theoremelab4g 3670* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   𝐵 = {𝑥𝜑}       (𝐴𝐵 ↔ (𝐴 ∈ V ∧ 𝜓))
 
Theoremelab3gf 3671 Membership in a class abstraction, with a weaker antecedent than elabgf 3663. (Contributed by NM, 6-Sep-2011.)
𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       ((𝜓𝐴𝐵) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
 
Theoremelab3g 3672* Membership in a class abstraction, with a weaker antecedent than elabg 3665. (Contributed by NM, 29-Aug-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝜓𝐴𝐵) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
 
Theoremelab3 3673* Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.)
(𝜓𝐴 ∈ V)    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴 ∈ {𝑥𝜑} ↔ 𝜓)
 
Theoremelrabi 3674* Implication for the membership in a restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
(𝐴 ∈ {𝑥𝑉𝜑} → 𝐴𝑉)
 
Theoremelrabf 3675 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 3676 Abstract builder using the constant wff (Contributed by Thierry Arnoux, 4-May-2020.)
𝑥𝐴       {𝑥𝐴 ∣ ⊤} = 𝐴
 
Theoremrabeqc 3677* 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 3678* Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 3680.) (Contributed by Thierry Arnoux, 31-Aug-2017.)
((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝜓))
 
Theoremelrab 3679* Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 21-May-1999.) Remove dependency on ax-13 2383. (Revised by Steven Nguyen, 23-Nov-2022.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵𝜓))
 
Theoremelrab3 3680* Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝐵 → (𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝜓))
 
Theoremelrabd 3681* Membership in a restricted class abstraction, using implicit substitution. Deduction version of elrab 3679. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝑥 = 𝐴 → (𝜓𝜒))    &   (𝜑𝐴𝐵)    &   (𝜑𝜒)       (𝜑𝐴 ∈ {𝑥𝐵𝜓})
 
Theoremelrab2 3682* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   𝐶 = {𝑥𝐵𝜑}       (𝐴𝐶 ↔ (𝐴𝐵𝜓))
 
Theoremralab 3683* Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝑦 = 𝑥 → (𝜑𝜓))       (∀𝑥 ∈ {𝑦𝜑}𝜒 ↔ ∀𝑥(𝜓𝜒))
 
Theoremralrab 3684* Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝑦 = 𝑥 → (𝜑𝜓))       (∀𝑥 ∈ {𝑦𝐴𝜑}𝜒 ↔ ∀𝑥𝐴 (𝜓𝜒))
 
Theoremrexab 3685* 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 3686* 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))
 
Theoremralab2 3687* Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) Drop ax-8 2107. (Revised by Gino Giotto, 1-Dec-2023.)
(𝑥 = 𝑦 → (𝜓𝜒))       (∀𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∀𝑦(𝜑𝜒))
 
Theoremralab2OLD 3688* Obsolete version of ralab2 3687 as of 1-Dec-2023. (Contributed by Mario Carneiro, 3-Sep-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜓𝜒))       (∀𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∀𝑦(𝜑𝜒))
 
Theoremralrab2 3689* Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝑥 = 𝑦 → (𝜓𝜒))       (∀𝑥 ∈ {𝑦𝐴𝜑}𝜓 ↔ ∀𝑦𝐴 (𝜑𝜒))
 
Theoremrexab2 3690* Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) Drop ax-8 2107. (Revised by Gino Giotto, 1-Dec-2023.)
(𝑥 = 𝑦 → (𝜓𝜒))       (∃𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∃𝑦(𝜑𝜒))
 
Theoremrexab2OLD 3691* Obsolete version of rexab2 3690 as of 1-Dec-2023. (Contributed by Mario Carneiro, 3-Sep-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜓𝜒))       (∃𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∃𝑦(𝜑𝜒))
 
Theoremrexrab2 3692* Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝑥 = 𝑦 → (𝜓𝜒))       (∃𝑥 ∈ {𝑦𝐴𝜑}𝜓 ↔ ∃𝑦𝐴 (𝜑𝜒))
 
Theoremabidnf 3693* Identity used to create closed-form versions of bound-variable hypothesis builders for class expressions. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Mario Carneiro, 12-Oct-2016.)
(𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
 
Theoremdedhb 3694* A deduction theorem for converting the inference 𝑥𝐴 => 𝜑 into a closed theorem. Use nfa1 2146 and nfab 2984 to eliminate the hypothesis of the substitution instance 𝜓 of the inference. For converting the inference form into a deduction form, abidnf 3693 is useful. (Contributed by NM, 8-Dec-2006.)
(𝐴 = {𝑧 ∣ ∀𝑥 𝑧𝐴} → (𝜑𝜓))    &   𝜓       (𝑥𝐴𝜑)
 
Theoremnelrdva 3695* Deduce negative membership from an implication. (Contributed by Thierry Arnoux, 27-Nov-2017.)
((𝜑𝑥𝐴) → 𝑥𝐵)       (𝜑 → ¬ 𝐵𝐴)
 
Theoremeqeu 3696* A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵𝜓 ∧ ∀𝑥(𝜑𝑥 = 𝐴)) → ∃!𝑥𝜑)
 
Theoremmoeq 3697* There exists at most one set equal to a given class. (Contributed by NM, 8-Mar-1995.) Shorten combined proofs of moeq 3697 and eueq 3698. (Proof shortened by BJ, 24-Sep-2022.)
∃*𝑥 𝑥 = 𝐴
 
Theoremeueq 3698* A class is a set if and only if there exists a unique set equal to it. (Contributed by NM, 25-Nov-1994.) Shorten combined proofs of moeq 3697 and eueq 3698. (Proof shortened by BJ, 24-Sep-2022.)
(𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
 
Theoremeueqi 3699* There exists a unique set equal to a given set. Inference associated with euequ 2679. See euequ 2679 in the case of a setvar. (Contributed by NM, 5-Apr-1995.)
𝐴 ∈ V       ∃!𝑥 𝑥 = 𝐴
 
Theoremeueq2 3700* Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.)
𝐴 ∈ V    &   𝐵 ∈ V       ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44804
  Copyright terms: Public domain < Previous  Next >