P. 30 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 2901-3000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rspc2va 2901*	2-variable restricted specialization, using implicit substitution. (Contributed by NM, 18-Jun-2014.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜓))    ⇒   ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜑) → 𝜓)
Theorem	rspc2ev 2902*	2-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 16-Oct-1999.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝜓) → ∃𝑥 ∈ 𝐶 ∃𝑦 ∈ 𝐷 𝜑)
Theorem	rspc3v 2903*	3-variable restricted specialization, using implicit substitution. (Contributed by NM, 10-May-2005.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑧 = 𝐶 → (𝜃 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 𝜑 → 𝜓))
Theorem	rspc3ev 2904*	3-variable restricted existentional specialization, using implicit substitution. (Contributed by NM, 25-Jul-2012.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑧 = 𝐶 → (𝜃 ↔ 𝜓))    ⇒   ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ 𝜓) → ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 ∃𝑧 ∈ 𝑇 𝜑)
Theorem	rspceeqv 2905*	Restricted existential specialization in an equality, using implicit substitution. (Contributed by BJ, 2-Sep-2022.)
⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)    ⇒   ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐸 = 𝐷) → ∃𝑥 ∈ 𝐵 𝐸 = 𝐶)
Theorem	eqvinc 2906*	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 = 𝐴 ∧ 𝑥 = 𝐵))
Theorem	eqvincg 2907*	A variable introduction law for class equality, deduction version. (Contributed by Thierry Arnoux, 2-Mar-2017.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵)))
Theorem	eqvincf 2908	A variable introduction law for class equality, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Sep-2003.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵))
Theorem	alexeq 2909*	Two ways to express substitution of 𝐴 for 𝑥 in 𝜑. (Contributed by NM, 2-Mar-1995.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
Theorem	ceqex 2910*	Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
Theorem	ceqsexg 2911*	A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 11-Oct-2004.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓))
Theorem	ceqsexgv 2912*	Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓))
Theorem	ceqsrexv 2913*	Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 30-Apr-2004.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝐵 → (∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓))
Theorem	ceqsrexbv 2914*	Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by Mario Carneiro, 14-Mar-2014.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))
Theorem	ceqsrex2v 2915*	Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 29-Oct-2005.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    ⇒   ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∃𝑥 ∈ 𝐶 ∃𝑦 ∈ 𝐷 ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝜑) ↔ 𝜒))
Theorem	clel2 2916*	An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
Theorem	clel3g 2917*	An alternate definition of class membership when the class is a set. (Contributed by NM, 13-Aug-2005.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥)))
Theorem	clel3 2918*	An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥))
Theorem	clel4 2919*	An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝑥))
Theorem	clel5 2920*	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.)
⊢ (𝑋 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑋 = 𝑥)
Theorem	pm13.183 2921*	Compare theorem *13.183 in [WhiteheadRussell] p. 178. Only 𝐴 is required to be a set. (Contributed by Andrew Salmon, 3-Jun-2011.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 ↔ ∀𝑧(𝑧 = 𝐴 ↔ 𝑧 = 𝐵)))
Theorem	rr19.3v 2922*	Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 25-Oct-2012.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)
Theorem	rr19.28v 2923*	Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 29-Oct-2012.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))
Theorem	elabgt 2924*	Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 2929.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
Theorem	elabgf 2925	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.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
Theorem	elabf 2926*	Membership in a class abstraction, using implicit substitution. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 12-Oct-2016.)
⊢ Ⅎ𝑥𝜓    &   ⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)
Theorem	elab 2927*	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 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)
Theorem	elabd 2928*	Explicit demonstration the class {𝑥 ∣ 𝜓} is not empty by the example 𝑋. (Contributed by RP, 12-Aug-2020.)
⊢ (𝜑 → 𝑋 ∈ V)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)
Theorem	elabg 2929*	Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
Theorem	elab2g 2930*	Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ 𝐵 = {𝑥 ∣ 𝜑}    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ 𝜓))
Theorem	elab2 2931*	Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ 𝐵 = {𝑥 ∣ 𝜑}    ⇒   ⊢ (𝐴 ∈ 𝐵 ↔ 𝜓)
Theorem	elab4g 2932*	Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ 𝐵 = {𝑥 ∣ 𝜑}    ⇒   ⊢ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ V ∧ 𝜓))
Theorem	elab3gf 2933	Membership in a class abstraction, with a weaker antecedent than elabgf 2925. (Contributed by NM, 6-Sep-2011.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝜓 → 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
Theorem	elab3g 2934*	Membership in a class abstraction, with a weaker antecedent than elabg 2929. (Contributed by NM, 29-Aug-2006.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝜓 → 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
Theorem	elab3 2935*	Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.)
⊢ (𝜓 → 𝐴 ∈ V)    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)
Theorem	elrabi 2936*	Implication for the membership in a restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
⊢ (𝐴 ∈ {𝑥 ∈ 𝑉 ∣ 𝜑} → 𝐴 ∈ 𝑉)
Theorem	elrabf 2937	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.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))
Theorem	elrab3t 2938*	Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 2940.) (Contributed by Thierry Arnoux, 31-Aug-2017.)
⊢ ((∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝜓))
Theorem	elrab 2939*	Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 21-May-1999.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))
Theorem	elrab3 2940*	Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝜓))
Theorem	elrabd 2941*	Membership in a restricted class abstraction, using implicit substitution. Deduction version of elrab 2939. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → 𝜒)    ⇒   ⊢ (𝜑 → 𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜓})
Theorem	elrab2 2942*	Membership in a class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ 𝜑}    ⇒   ⊢ (𝐴 ∈ 𝐶 ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))
Theorem	ralab 2943*	Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∀𝑥(𝜓 → 𝜒))
Theorem	ralrab 2944*	Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜒 ↔ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
Theorem	rexab 2945*	Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.)
⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∃𝑥(𝜓 ∧ 𝜒))
Theorem	rexrab 2946*	Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.)
⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜒 ↔ ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒))
Theorem	ralab2 2947*	Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∀𝑦(𝜑 → 𝜒))
Theorem	ralrab2 2948*	Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (∀𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜓 ↔ ∀𝑦 ∈ 𝐴 (𝜑 → 𝜒))
Theorem	rexab2 2949*	Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∃𝑦(𝜑 ∧ 𝜒))
Theorem	rexrab2 2950*	Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (∃𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜓 ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝜒))
Theorem	abidnf 2951*	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.)
⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴)
Theorem	dedhb 2952*	A deduction theorem for converting the inference ⊢ Ⅎ𝑥𝐴 => ⊢ 𝜑 into a closed theorem. Use nfa1 1567 and nfab 2357 to eliminate the hypothesis of the substitution instance 𝜓 of the inference. For converting the inference form into a deduction form, abidnf 2951 is useful. (Contributed by NM, 8-Dec-2006.)
⊢ (𝐴 = {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} → (𝜑 ↔ 𝜓))    &   ⊢ 𝜓    ⇒   ⊢ (Ⅎ𝑥𝐴 → 𝜑)
Theorem	eqeu 2953*	A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓 ∧ ∀𝑥(𝜑 → 𝑥 = 𝐴)) → ∃!𝑥𝜑)
Theorem	eueq 2954*	Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.)
⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
Theorem	eueq1 2955*	Equality has existential uniqueness. (Contributed by NM, 5-Apr-1995.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∃!𝑥 𝑥 = 𝐴
Theorem	eueq2dc 2956*	Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (DECID 𝜑 → ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵)))
Theorem	eueq3dc 2957*	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    &   ⊢ ¬ (𝜑 ∧ 𝜓)    ⇒   ⊢ (DECID 𝜑 → (DECID 𝜓 → ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶))))
Theorem	moeq 2958*	There is at most one set equal to a class. (Contributed by NM, 8-Mar-1995.)
⊢ ∃*𝑥 𝑥 = 𝐴
Theorem	moeq3dc 2959*	"At most one" property of equality (split into 3 cases). (Contributed by Jim Kingdon, 7-Jul-2018.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    &   ⊢ ¬ (𝜑 ∧ 𝜓)    ⇒   ⊢ (DECID 𝜑 → (DECID 𝜓 → ∃*𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶))))
Theorem	mosubt 2960*	"At most one" remains true after substitution. (Contributed by Jim Kingdon, 18-Jan-2019.)
⊢ (∀𝑦∃*𝑥𝜑 → ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑))
Theorem	mosub 2961*	"At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995.)
⊢ ∃*𝑥𝜑    ⇒   ⊢ ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑)
Theorem	mo2icl 2962*	Theorem for inferring "at most one". (Contributed by NM, 17-Oct-1996.)
⊢ (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑)
Theorem	mob2 2963*	Consequence of "at most one". (Contributed by NM, 2-Jan-2015.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑 ∧ 𝜑) → (𝑥 = 𝐴 ↔ 𝜓))
Theorem	moi2 2964*	Consequence of "at most one". (Contributed by NM, 29-Jun-2008.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑) ∧ (𝜑 ∧ 𝜓)) → 𝑥 = 𝐴)
Theorem	mob 2965*	Equality implied by "at most one". (Contributed by NM, 18-Feb-2006.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    ⇒   ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∃*𝑥𝜑 ∧ 𝜓) → (𝐴 = 𝐵 ↔ 𝜒))
Theorem	moi 2966*	Equality implied by "at most one". (Contributed by NM, 18-Feb-2006.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    ⇒   ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∃*𝑥𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝐴 = 𝐵)
Theorem	morex 2967*	Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ 𝐵 ∈ V    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥𝜑) → (𝜓 → 𝐵 ∈ 𝐴))
Theorem	euxfr2dc 2968*	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    ⇒   ⊢ (DECID ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦𝜑))
Theorem	euxfrdc 2969*	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 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (DECID ∃𝑦∃𝑥(𝑥 = 𝐴 ∧ 𝜓) → (∃!𝑥𝜑 ↔ ∃!𝑦𝜓))
Theorem	euind 2970*	Existential uniqueness via an indirect equality. (Contributed by NM, 11-Oct-2010.)
⊢ 𝐵 ∈ V    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   ⊢ ((∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) ∧ ∃𝑥𝜑) → ∃!𝑧∀𝑥(𝜑 → 𝑧 = 𝐴))
Theorem	reu2 2971*	A way to express restricted uniqueness. (Contributed by NM, 22-Nov-1994.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
Theorem	reu6 2972*	A way to express restricted uniqueness. (Contributed by NM, 20-Oct-2006.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦))
Theorem	reu3 2973*	A way to express restricted uniqueness. (Contributed by NM, 24-Oct-2006.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦)))
Theorem	reu6i 2974*	A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ ((𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵)) → ∃!𝑥 ∈ 𝐴 𝜑)
Theorem	eqreu 2975*	A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐵 ∈ 𝐴 ∧ 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵)) → ∃!𝑥 ∈ 𝐴 𝜑)
Theorem	rmo4 2976*	Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by NM, 16-Jun-2017.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
Theorem	reu4 2977*	Restricted uniqueness using implicit substitution. (Contributed by NM, 23-Nov-1994.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
Theorem	reu7 2978*	Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
Theorem	reu8 2979*	Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
Theorem	rmo3f 2980*	Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
Theorem	rmo4f 2981*	Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by Thierry Arnoux, 11-Oct-2016.) (Revised by Thierry Arnoux, 8-Mar-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
Theorem	reueq 2982*	Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.)
⊢ (𝐵 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 𝑥 = 𝐵)
Theorem	rmoan 2983	Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.)
⊢ (∃*𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 (𝜓 ∧ 𝜑))
Theorem	rmoim 2984	Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑))
Theorem	rmoimia 2985	Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑)
Theorem	rmoimi2 2986	Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜓))    ⇒   ⊢ (∃*𝑥 ∈ 𝐵 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑)
Theorem	2reuswapdc 2987*	A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by NM, 16-Jun-2017.)
⊢ (DECID ∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) → (∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 → (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)))
Theorem	reuind 2988*	Existential uniqueness via an indirect equality. (Contributed by NM, 16-Oct-2010.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   ⊢ ((∀𝑥∀𝑦(((𝐴 ∈ 𝐶 ∧ 𝜑) ∧ (𝐵 ∈ 𝐶 ∧ 𝜓)) → 𝐴 = 𝐵) ∧ ∃𝑥(𝐴 ∈ 𝐶 ∧ 𝜑)) → ∃!𝑧 ∈ 𝐶 ∀𝑥((𝐴 ∈ 𝐶 ∧ 𝜑) → 𝑧 = 𝐴))
Theorem	2rmorex 2989*	Double restricted quantification with "at most one," analogous to 2moex 2144. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝜑)
Theorem	nelrdva 2990*	Deduce negative membership from an implication. (Contributed by Thierry Arnoux, 27-Nov-2017.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ 𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐴)
2.1.7  Conditional equality (experimental) This is a very useless definition, which "abbreviates" (𝑥 = 𝑦 → 𝜑) as CondEq(𝑥 = 𝑦 → 𝜑). What this display hides, though, is that the first expression, even though it has a shorter constant string, is actually much more complicated in its parse tree: it is parsed as (wi (wceq (cv vx) (cv vy)) wph), while the CondEq version is parsed as (wcdeq vx vy wph). It also allows us to give a name to the specific ternary operation (𝑥 = 𝑦 → 𝜑). This is all used as part of a metatheorem: we want to say that ⊢ (𝑥 = 𝑦 → (𝜑(𝑥) ↔ 𝜑(𝑦))) and ⊢ (𝑥 = 𝑦 → 𝐴(𝑥) = 𝐴(𝑦)) are provable, for any expressions 𝜑(𝑥) or 𝐴(𝑥) in the language. The proof is by induction, so the base case is each of the primitives, which is why you will see a theorem for each of the set.mm primitive operations. The metatheorem comes with a disjoint variables condition: every variable in 𝜑(𝑥) is assumed disjoint from 𝑥 except 𝑥 itself. For such a proof by induction, we must consider each of the possible forms of 𝜑(𝑥). If it is a variable other than 𝑥, then we have CondEq(𝑥 = 𝑦 → 𝐴 = 𝐴) or CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜑)), which is provable by cdeqth 2995 and reflexivity. Since we are only working with class and wff expressions, it can't be 𝑥 itself in set.mm, but if it was we'd have to also prove CondEq(𝑥 = 𝑦 → 𝑥 = 𝑦) (where set equality is being used on the right). Otherwise, it is a primitive operation applied to smaller expressions. In these cases, for each setvar variable parameter to the operation, we must consider if it is equal to 𝑥 or not, which yields 2^n proof obligations. Luckily, all primitive operations in set.mm have either zero or one set variable, so we only need to prove one statement for the non-set constructors (like implication) and two for the constructors taking a set (the forall and the class builder). In each of the primitive proofs, we are allowed to assume that 𝑦 is disjoint from 𝜑(𝑥) and vice versa, because this is maintained through the induction. This is how we satisfy the disjoint variable conditions of cdeqab1 3000 and cdeqab 2998.
Syntax	wcdeq 2991	Extend wff notation to include conditional equality. This is a technical device used in the proof that Ⅎ is the not-free predicate, and that definitions are conservative as a result.
wff CondEq(𝑥 = 𝑦 → 𝜑)
Definition	df-cdeq 2992	Define conditional equality. All the notation to the left of the ↔ is fake; the parentheses and arrows are all part of the notation, which could equally well be written CondEq𝑥𝑦𝜑. On the right side is the actual implication arrow. The reason for this definition is to "flatten" the structure on the right side (whose tree structure is something like (wi (wceq (cv vx) (cv vy)) wph) ) into just (wcdeq vx vy wph). (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (CondEq(𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑))
Theorem	cdeqi 2993	Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ CondEq(𝑥 = 𝑦 → 𝜑)
Theorem	cdeqri 2994	Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ CondEq(𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ (𝑥 = 𝑦 → 𝜑)
Theorem	cdeqth 2995	Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ 𝜑    ⇒   ⊢ CondEq(𝑥 = 𝑦 → 𝜑)
Theorem	cdeqnot 2996	Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ CondEq(𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
Theorem	cdeqal 2997*	Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ CondEq(𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))
Theorem	cdeqab 2998*	Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ CondEq(𝑥 = 𝑦 → {𝑧 ∣ 𝜑} = {𝑧 ∣ 𝜓})
Theorem	cdeqal1 2999*	Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ CondEq(𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
Theorem	cdeqab1 3000*	Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ CondEq(𝑥 = 𝑦 → {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓})