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Theorem List for Intuitionistic Logic Explorer - 2801-2900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremspcedv 2801* Existential specialization, using implicit substitution, deduction version. (Contributed by RP, 12-Aug-2020.)
(𝜑𝑋 ∈ V)    &   (𝜑𝜒)    &   (𝑥 = 𝑋 → (𝜓𝜒))       (𝜑 → ∃𝑥𝜓)
 
Theoremspc2egv 2802* Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊) → (𝜓 → ∃𝑥𝑦𝜑))
 
Theoremspc2gv 2803* Specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 27-Apr-2004.)
((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊) → (∀𝑥𝑦𝜑𝜓))
 
Theoremspc3egv 2804* Existential specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)
((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝜓 → ∃𝑥𝑦𝑧𝜑))
 
Theoremspc3gv 2805* Specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)
((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∀𝑥𝑦𝑧𝜑𝜓))
 
Theoremspcv 2806* Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥𝜑𝜓)
 
Theoremspcev 2807* 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 2808* Existential specialization, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
𝐴 ∈ V    &   𝐵 ∈ V    &   ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       (𝜓 → ∃𝑥𝑦𝜑)
 
Theoremrspct 2809* A closed version of rspc 2810. (Contributed by Andrew Salmon, 6-Jun-2011.)
𝑥𝜓       (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓)))
 
Theoremrspc 2810* 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 2811* 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)
 
Theoremrspcv 2812* Restricted specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))
 
Theoremrspccv 2813* Restricted specialization, using implicit substitution. (Contributed by NM, 2-Feb-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥𝐵 𝜑 → (𝐴𝐵𝜓))
 
Theoremrspcva 2814* Restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-2005.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵 ∧ ∀𝑥𝐵 𝜑) → 𝜓)
 
Theoremrspccva 2815* 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 2816* Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)
 
Theoremrspcimdv 2817* Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐵 𝜓𝜒))
 
Theoremrspcimedv 2818* Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜒𝜓))       (𝜑 → (𝜒 → ∃𝑥𝐵 𝜓))
 
Theoremrspcdv 2819* 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 2820* 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))
 
Theoremrspcdva 2821* Restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 21-Jun-2020.)
(𝑥 = 𝐶 → (𝜓𝜒))    &   (𝜑 → ∀𝑥𝐴 𝜓)    &   (𝜑𝐶𝐴)       (𝜑𝜒)
 
Theoremrspcedvd 2822* Restricted existential specialization, using implicit substitution. Variant of rspcedv 2820. (Contributed by AV, 27-Nov-2019.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))    &   (𝜑𝜒)       (𝜑 → ∃𝑥𝐵 𝜓)
 
Theoremrspcime 2823* Prove a restricted existential. (Contributed by Rohan Ridenour, 3-Aug-2023.)
((𝜑𝑥 = 𝐴) → 𝜓)    &   (𝜑𝐴𝐵)       (𝜑 → ∃𝑥𝐵 𝜓)
 
Theoremrspceaimv 2824* Restricted existential specialization of a universally quantified implication. (Contributed by BJ, 24-Aug-2022.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵 ∧ ∀𝑦𝐶 (𝜓𝜒)) → ∃𝑥𝐵𝑦𝐶 (𝜑𝜒))
 
Theoremrspcedeq1vd 2825* Restricted existential specialization, using implicit substitution. Variant of rspcedvd 2822 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 2826* Restricted existential specialization, using implicit substitution. Variant of rspcedvd 2822 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 2827* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 9-Nov-2012.)
𝑥𝜒    &   𝑦𝜓    &   (𝑥 = 𝐴 → (𝜑𝜒))    &   (𝑦 = 𝐵 → (𝜒𝜓))       ((𝐴𝐶𝐵𝐷) → (∀𝑥𝐶𝑦𝐷 𝜑𝜓))
 
Theoremrspc2gv 2828* Restricted specialization with two quantifiers, using implicit substitution. (Contributed by BJ, 2-Dec-2021.)
((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊) → (∀𝑥𝑉𝑦𝑊 𝜑𝜓))
 
Theoremrspc2v 2829* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-1999.)
(𝑥 = 𝐴 → (𝜑𝜒))    &   (𝑦 = 𝐵 → (𝜒𝜓))       ((𝐴𝐶𝐵𝐷) → (∀𝑥𝐶𝑦𝐷 𝜑𝜓))
 
Theoremrspc2va 2830* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 18-Jun-2014.)
(𝑥 = 𝐴 → (𝜑𝜒))    &   (𝑦 = 𝐵 → (𝜒𝜓))       (((𝐴𝐶𝐵𝐷) ∧ ∀𝑥𝐶𝑦𝐷 𝜑) → 𝜓)
 
Theoremrspc2ev 2831* 2-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 16-Oct-1999.)
(𝑥 = 𝐴 → (𝜑𝜒))    &   (𝑦 = 𝐵 → (𝜒𝜓))       ((𝐴𝐶𝐵𝐷𝜓) → ∃𝑥𝐶𝑦𝐷 𝜑)
 
Theoremrspc3v 2832* 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 2833* 3-variable restricted existentional specialization, using implicit substitution. (Contributed by NM, 25-Jul-2012.)
(𝑥 = 𝐴 → (𝜑𝜒))    &   (𝑦 = 𝐵 → (𝜒𝜃))    &   (𝑧 = 𝐶 → (𝜃𝜓))       (((𝐴𝑅𝐵𝑆𝐶𝑇) ∧ 𝜓) → ∃𝑥𝑅𝑦𝑆𝑧𝑇 𝜑)
 
Theoremrspceeqv 2834* Restricted existential specialization in an equality, using implicit substitution. (Contributed by BJ, 2-Sep-2022.)
(𝑥 = 𝐴𝐶 = 𝐷)       ((𝐴𝐵𝐸 = 𝐷) → ∃𝑥𝐵 𝐸 = 𝐶)
 
Theoremeqvinc 2835* A variable introduction law for class equality. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
𝐴 ∈ V       (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵))
 
Theoremeqvincg 2836* A variable introduction law for class equality, deduction version. (Contributed by Thierry Arnoux, 2-Mar-2017.)
(𝐴𝑉 → (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵)))
 
Theoremeqvincf 2837 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))
 
Theoremalexeq 2838* Two ways to express substitution of 𝐴 for 𝑥 in 𝜑. (Contributed by NM, 2-Mar-1995.)
𝐴 ∈ V       (∀𝑥(𝑥 = 𝐴𝜑) ↔ ∃𝑥(𝑥 = 𝐴𝜑))
 
Theoremceqex 2839* Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.)
(𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
 
Theoremceqsexg 2840* 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 2841* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
 
Theoremceqsrexv 2842* Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 30-Apr-2004.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝐵 → (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
 
Theoremceqsrexbv 2843* Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by Mario Carneiro, 14-Mar-2014.)
(𝑥 = 𝐴 → (𝜑𝜓))       (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ (𝐴𝐵𝜓))
 
Theoremceqsrex2v 2844* 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))
 
Theoremclel2 2845* 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 2846* An alternate definition of class membership when the class is a set. (Contributed by NM, 13-Aug-2005.)
(𝐵𝑉 → (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐵𝐴𝑥)))
 
Theoremclel3 2847* 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 2848* 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 2849* 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.) (Revised by Steven Nguyen, 19-May-2023.)
(𝑋𝐴 ↔ ∃𝑥𝐴 𝑋 = 𝑥)
 
Theorempm13.183 2850* Compare theorem *13.183 in [WhiteheadRussell] p. 178. Only 𝐴 is required to be a set. (Contributed by Andrew Salmon, 3-Jun-2011.)
(𝐴𝑉 → (𝐴 = 𝐵 ↔ ∀𝑧(𝑧 = 𝐴𝑧 = 𝐵)))
 
Theoremrr19.3v 2851* Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 25-Oct-2012.)
(∀𝑥𝐴𝑦𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑)
 
Theoremrr19.28v 2852* Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 29-Oct-2012.)
(∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓))
 
Theoremelabgt 2853* Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 2858.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
((𝐴𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
 
Theoremelabgf 2854 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 2855* 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)
 
Theoremelab 2856* 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 2857* 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)
 
Theoremelabg 2858* Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
 
Theoremelab2g 2859* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   𝐵 = {𝑥𝜑}       (𝐴𝑉 → (𝐴𝐵𝜓))
 
Theoremelab2 2860* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝐵 = {𝑥𝜑}       (𝐴𝐵𝜓)
 
Theoremelab4g 2861* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   𝐵 = {𝑥𝜑}       (𝐴𝐵 ↔ (𝐴 ∈ V ∧ 𝜓))
 
Theoremelab3gf 2862 Membership in a class abstraction, with a weaker antecedent than elabgf 2854. (Contributed by NM, 6-Sep-2011.)
𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       ((𝜓𝐴𝐵) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
 
Theoremelab3g 2863* Membership in a class abstraction, with a weaker antecedent than elabg 2858. (Contributed by NM, 29-Aug-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝜓𝐴𝐵) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
 
Theoremelab3 2864* Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.)
(𝜓𝐴 ∈ V)    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴 ∈ {𝑥𝜑} ↔ 𝜓)
 
Theoremelrabi 2865* Implication for the membership in a restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
(𝐴 ∈ {𝑥𝑉𝜑} → 𝐴𝑉)
 
Theoremelrabf 2866 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))
 
Theoremelrab3t 2867* Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 2869.) (Contributed by Thierry Arnoux, 31-Aug-2017.)
((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝜓))
 
Theoremelrab 2868* Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 21-May-1999.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵𝜓))
 
Theoremelrab3 2869* Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝐵 → (𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝜓))
 
Theoremelrabd 2870* Membership in a restricted class abstraction, using implicit substitution. Deduction version of elrab 2868. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝑥 = 𝐴 → (𝜓𝜒))    &   (𝜑𝐴𝐵)    &   (𝜑𝜒)       (𝜑𝐴 ∈ {𝑥𝐵𝜓})
 
Theoremelrab2 2871* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   𝐶 = {𝑥𝐵𝜑}       (𝐴𝐶 ↔ (𝐴𝐵𝜓))
 
Theoremralab 2872* Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝑦 = 𝑥 → (𝜑𝜓))       (∀𝑥 ∈ {𝑦𝜑}𝜒 ↔ ∀𝑥(𝜓𝜒))
 
Theoremralrab 2873* Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝑦 = 𝑥 → (𝜑𝜓))       (∀𝑥 ∈ {𝑦𝐴𝜑}𝜒 ↔ ∀𝑥𝐴 (𝜓𝜒))
 
Theoremrexab 2874* 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 2875* 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 2876* Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝑥 = 𝑦 → (𝜓𝜒))       (∀𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∀𝑦(𝜑𝜒))
 
Theoremralrab2 2877* Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝑥 = 𝑦 → (𝜓𝜒))       (∀𝑥 ∈ {𝑦𝐴𝜑}𝜓 ↔ ∀𝑦𝐴 (𝜑𝜒))
 
Theoremrexab2 2878* Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝑥 = 𝑦 → (𝜓𝜒))       (∃𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∃𝑦(𝜑𝜒))
 
Theoremrexrab2 2879* Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝑥 = 𝑦 → (𝜓𝜒))       (∃𝑥 ∈ {𝑦𝐴𝜑}𝜓 ↔ ∃𝑦𝐴 (𝜑𝜒))
 
Theoremabidnf 2880* 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 2881* A deduction theorem for converting the inference 𝑥𝐴 => 𝜑 into a closed theorem. Use nfa1 1521 and nfab 2304 to eliminate the hypothesis of the substitution instance 𝜓 of the inference. For converting the inference form into a deduction form, abidnf 2880 is useful. (Contributed by NM, 8-Dec-2006.)
(𝐴 = {𝑧 ∣ ∀𝑥 𝑧𝐴} → (𝜑𝜓))    &   𝜓       (𝑥𝐴𝜑)
 
Theoremeqeu 2882* A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵𝜓 ∧ ∀𝑥(𝜑𝑥 = 𝐴)) → ∃!𝑥𝜑)
 
Theoremeueq 2883* Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.)
(𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
 
Theoremeueq1 2884* Equality has existential uniqueness. (Contributed by NM, 5-Apr-1995.)
𝐴 ∈ V       ∃!𝑥 𝑥 = 𝐴
 
Theoremeueq2dc 2885* Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.)
𝐴 ∈ V    &   𝐵 ∈ V       (DECID 𝜑 → ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ 𝜑𝑥 = 𝐵)))
 
Theoremeueq3dc 2886* Equality has existential uniqueness (split into 3 cases). (Contributed by NM, 5-Apr-1995.) (Proof shortened by Mario Carneiro, 28-Sep-2015.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &    ¬ (𝜑𝜓)       (DECID 𝜑 → (DECID 𝜓 → ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶))))
 
Theoremmoeq 2887* There is at most one set equal to a class. (Contributed by NM, 8-Mar-1995.)
∃*𝑥 𝑥 = 𝐴
 
Theoremmoeq3dc 2888* "At most one" property of equality (split into 3 cases). (Contributed by Jim Kingdon, 7-Jul-2018.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &    ¬ (𝜑𝜓)       (DECID 𝜑 → (DECID 𝜓 → ∃*𝑥((𝜑𝑥 = 𝐴) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶))))
 
Theoremmosubt 2889* "At most one" remains true after substitution. (Contributed by Jim Kingdon, 18-Jan-2019.)
(∀𝑦∃*𝑥𝜑 → ∃*𝑥𝑦(𝑦 = 𝐴𝜑))
 
Theoremmosub 2890* "At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995.)
∃*𝑥𝜑       ∃*𝑥𝑦(𝑦 = 𝐴𝜑)
 
Theoremmo2icl 2891* Theorem for inferring "at most one." (Contributed by NM, 17-Oct-1996.)
(∀𝑥(𝜑𝑥 = 𝐴) → ∃*𝑥𝜑)
 
Theoremmob2 2892* Consequence of "at most one." (Contributed by NM, 2-Jan-2015.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵 ∧ ∃*𝑥𝜑𝜑) → (𝑥 = 𝐴𝜓))
 
Theoremmoi2 2893* Consequence of "at most one." (Contributed by NM, 29-Jun-2008.)
(𝑥 = 𝐴 → (𝜑𝜓))       (((𝐴𝐵 ∧ ∃*𝑥𝜑) ∧ (𝜑𝜓)) → 𝑥 = 𝐴)
 
Theoremmob 2894* Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))       (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑𝜓) → (𝐴 = 𝐵𝜒))
 
Theoremmoi 2895* Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))       (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑 ∧ (𝜓𝜒)) → 𝐴 = 𝐵)
 
Theoremmorex 2896* Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝐵 ∈ V    &   (𝑥 = 𝐵 → (𝜑𝜓))       ((∃𝑥𝐴 𝜑 ∧ ∃*𝑥𝜑) → (𝜓𝐵𝐴))
 
Theoremeuxfr2dc 2897* Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 14-Nov-2004.)
𝐴 ∈ V    &   ∃*𝑦 𝑥 = 𝐴       (DECID𝑦𝑥(𝑥 = 𝐴𝜑) → (∃!𝑥𝑦(𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝜑))
 
Theoremeuxfrdc 2898* Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 14-Nov-2004.)
𝐴 ∈ V    &   ∃!𝑦 𝑥 = 𝐴    &   (𝑥 = 𝐴 → (𝜑𝜓))       (DECID𝑦𝑥(𝑥 = 𝐴𝜓) → (∃!𝑥𝜑 ↔ ∃!𝑦𝜓))
 
Theoremeuind 2899* Existential uniqueness via an indirect equality. (Contributed by NM, 11-Oct-2010.)
𝐵 ∈ V    &   (𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 = 𝑦𝐴 = 𝐵)       ((∀𝑥𝑦((𝜑𝜓) → 𝐴 = 𝐵) ∧ ∃𝑥𝜑) → ∃!𝑧𝑥(𝜑𝑧 = 𝐴))
 
Theoremreu2 2900* A way to express restricted uniqueness. (Contributed by NM, 22-Nov-1994.)
(∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
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