P. 29 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 2801-2900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	3eqtr4g 2801	A chained equality inference, useful for converting to definitions. (Contributed by NM, 21-Jun-1993.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ 𝐶 = 𝐴    &   ⊢ 𝐷 = 𝐵    ⇒   ⊢ (𝜑 → 𝐶 = 𝐷)
Theorem	3eqtr4a 2802	A chained equality inference, useful for converting to definitions. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜑 → 𝐶 = 𝐴)    &   ⊢ (𝜑 → 𝐷 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐶 = 𝐷)
Theorem	eq2tri 2803	A compound transitive inference for class equality. (Contributed by NM, 22-Jan-2004.)
⊢ (𝐴 = 𝐶 → 𝐷 = 𝐹)    &   ⊢ (𝐵 = 𝐷 → 𝐶 = 𝐺)    ⇒   ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐹) ↔ (𝐵 = 𝐷 ∧ 𝐴 = 𝐺))
Theorem	abbi1 2804	Equivalent formulas yield equal class abstractions (closed form). This is the forward implication of abbi 2808, proved from fewer axioms. (Contributed by BJ and WL and SN, 20-Aug-2023.)
⊢ (∀𝑥(𝜑 ↔ 𝜓) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓})
Theorem	abbidv 2805*	Equivalent wff's yield equal class abstractions (deduction form). (Contributed by NM, 10-Aug-1993.) Avoid ax-12 2170, based on an idea of Steven Nguyen. (Revised by Wolf Lammen, 6-May-2023.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒})
Theorem	abbii 2806	Equivalent wff's yield equal class abstractions (inference form). (Contributed by NM, 26-May-1993.) Remove dependency on ax-10 2136, ax-11 2153, and ax-12 2170. (Revised by Steven Nguyen, 3-May-2023.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓}
Theorem	abbid 2807	Equivalent wff's yield equal class abstractions (deduction form, with nonfreeness hypothesis). (Contributed by NM, 21-Jun-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) Avoid ax-10 2136 and ax-11 2153. (Revised by Wolf Lammen, 6-May-2023.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒})
Theorem	abbi 2808	Equivalent formulas define equal class abstractions, and conversely. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-8 2107 and df-clel 2814 (by avoiding use of cleqh 2860). (Revised by BJ, 23-Jun-2019.)
⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓})
Theorem	cbvabv 2809*	Rule used to change bound variables, using implicit substitution. Version of cbvab 2812 with disjoint variable conditions requiring fewer axioms. (Contributed by NM, 26-May-1999.) Require 𝑥, 𝑦 be disjoint to avoid ax-11 2153 and ax-13 2370. (Revised by Steven Nguyen, 4-Dec-2022.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}
Theorem	cbvabw 2810*	Rule used to change bound variables, using implicit substitution. Version of cbvab 2812 with a disjoint variable condition, which does not require ax-10 2136, ax-13 2370. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Gino Giotto, 23-May-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}
Theorem	cbvabwOLD 2811*	Obsolete version of cbvabw 2810 as of 23-May-2024. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}
Theorem	cbvab 2812	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2370. Usage of the weaker cbvabw 2810 and cbvabv 2809 are preferred. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}
Theorem	abeq2w 2813*	Version of abeq2 2870 using implicit substitution, which requires fewer axioms. (Contributed by GG and AV, 18-Sep-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ 𝜓))
2.1.2.3  Class membership
Definition	df-clel 2814*	Define the membership connective between classes. Theorem 6.3 of [Quine] p. 41, or Proposition 4.6 of [TakeutiZaring] p. 13, which we adopt as a definition. See these references for its metalogical justification. The hypotheses express that all instances of the conclusion where class variables are replaced with setvar variables hold. Therefore, this definition merely extends to class variables something that is true for setvar variables, hence is conservative. This is only a proof sketch of conservativity; for details see Appendix of [Levy] p. 357. This is the reason why we call this axiomatic statement a "definition", even though it does not have the usual form of a definition. If we required a definition to have the usual form, we would call df-clel 2814 an axiom. See also comments under df-clab 2714, df-cleq 2728, and abeq2 2870. Alternate characterizations of 𝐴 ∈ 𝐵 when either 𝐴 or 𝐵 is a set are given by clel2g 3598, clel3g 3601, and clel4g 3603. This is called the "axiom of membership" by [Levy] p. 338, who treats the theory of classes as an extralogical extension to our logic and set theory axioms. While the three class definitions df-clab 2714, df-cleq 2728, and df-clel 2814 are eliminable and conservative and thus meet the requirements for sound definitions, they are technically axioms in that they do not satisfy the requirements for the current definition checker. The proofs of conservativity require external justification that is beyond the scope of the definition checker. For a general discussion of the theory of classes, see mmset.html#class 2814. (Contributed by NM, 26-May-1993.) (Revised by BJ, 27-Jun-2019.)
⊢ (𝑦 ∈ 𝑧 ↔ ∃𝑢(𝑢 = 𝑦 ∧ 𝑢 ∈ 𝑧))    &   ⊢ (𝑡 ∈ 𝑡 ↔ ∃𝑣(𝑣 = 𝑡 ∧ 𝑣 ∈ 𝑡))    ⇒   ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
Theorem	dfclel 2815*	Characterization of the elements of a class. (Contributed by BJ, 27-Jun-2019.)
⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
Theorem	elex2 2816*	If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.) Avoid ax-9 2115, ax-ext 2707, df-clab 2714. (Revised by Wolf Lammen, 30-Nov-2024.)
⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵)
Theorem	elissetv 2817*	An element of a class exists. Version of elisset 2818 with a disjoint variable condition on 𝑉, 𝑥, avoiding df-clab 2714. Prefer its use over elisset 2818 when sufficient (for instance in usages where 𝑥 is a dummy variable). (Contributed by BJ, 14-Sep-2019.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
Theorem	elisset 2818*	An element of a class exists. Use elissetv 2817 instead when sufficient (for instance in usages where 𝑥 is a dummy variable). (Contributed by NM, 1-May-1995.) Reduce dependencies on axioms. (Revised by BJ, 29-Apr-2019.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
Theorem	eleq1w 2819	Weaker version of eleq1 2824 (but more general than elequ1 2112) not depending on ax-ext 2707 nor df-cleq 2728. Note that this provides a proof of ax-8 2107 from Tarski's FOL and dfclel 2815 (simply consider an instance where 𝐴 is replaced by a setvar and deduce the forward implication by biimpd 228), which shows that dfclel 2815 is too powerful to be used as a definition instead of df-clel 2814. (Contributed by BJ, 24-Jun-2019.)
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
Theorem	eleq2w 2820	Weaker version of eleq2 2825 (but more general than elequ2 2120) not depending on ax-ext 2707 nor df-cleq 2728. (Contributed by BJ, 29-Sep-2019.)
⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
Theorem	eleq1d 2821	Deduction from equality to equivalence of membership. (Contributed by NM, 21-Jun-1993.) Allow shortening of eleq1 2824. (Revised by Wolf Lammen, 20-Nov-2019.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
Theorem	eleq2d 2822	Deduction from equality to equivalence of membership. (Contributed by NM, 27-Dec-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 5-Dec-2019.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵))
Theorem	eleq2dALT 2823	Alternate proof of eleq2d 2822, shorter at the expense of requiring ax-12 2170. (Contributed by NM, 27-Dec-1993.) (Revised by Wolf Lammen, 20-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵))
Theorem	eleq1 2824	Equality implies equivalence of membership. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 20-Nov-2019.)
⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
Theorem	eleq2 2825	Equality implies equivalence of membership. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 20-Nov-2019.)
⊢ (𝐴 = 𝐵 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵))
Theorem	eleq12 2826	Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷))
Theorem	eleq1i 2827	Inference from equality to equivalence of membership. (Contributed by NM, 21-Jun-1993.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)
Theorem	eleq2i 2828	Inference from equality to equivalence of membership. (Contributed by NM, 26-May-1993.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵)
Theorem	eleq12i 2829	Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)
Theorem	eqneltri 2830	If a class is not an element of another class, an equal class is also not an element. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ 𝐴 = 𝐵    &   ⊢ ¬ 𝐵 ∈ 𝐶    ⇒   ⊢ ¬ 𝐴 ∈ 𝐶
Theorem	eleq12d 2831	Deduction from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷))
Theorem	eleq1a 2832	A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.)
⊢ (𝐴 ∈ 𝐵 → (𝐶 = 𝐴 → 𝐶 ∈ 𝐵))
Theorem	eqeltri 2833	Substitution of equal classes into membership relation. (Contributed by NM, 21-Jun-1993.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐵 ∈ 𝐶    ⇒   ⊢ 𝐴 ∈ 𝐶
Theorem	eqeltrri 2834	Substitution of equal classes into membership relation. (Contributed by NM, 21-Jun-1993.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐴 ∈ 𝐶    ⇒   ⊢ 𝐵 ∈ 𝐶
Theorem	eleqtri 2835	Substitution of equal classes into membership relation. (Contributed by NM, 15-Jul-1993.)
⊢ 𝐴 ∈ 𝐵    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ 𝐴 ∈ 𝐶
Theorem	eleqtrri 2836	Substitution of equal classes into membership relation. (Contributed by NM, 15-Jul-1993.)
⊢ 𝐴 ∈ 𝐵    &   ⊢ 𝐶 = 𝐵    ⇒   ⊢ 𝐴 ∈ 𝐶
Theorem	eqeltrd 2837	Substitution of equal classes into membership relation, deduction form. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	eqeltrrd 2838	Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐵 ∈ 𝐶)
Theorem	eleqtrd 2839	Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	eleqtrrd 2840	Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	eqeltrid 2841	A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜑 → 𝐵 ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	eqeltrrid 2842	A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
⊢ 𝐵 = 𝐴    &   ⊢ (𝜑 → 𝐵 ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	eleqtrid 2843	A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
⊢ 𝐴 ∈ 𝐵    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	eleqtrrid 2844	A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
⊢ 𝐴 ∈ 𝐵    &   ⊢ (𝜑 → 𝐶 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	eqeltrdi 2845	A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ 𝐵 ∈ 𝐶    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	eqeltrrdi 2846	A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
⊢ (𝜑 → 𝐵 = 𝐴)    &   ⊢ 𝐵 ∈ 𝐶    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	eleqtrdi 2847	A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	eleqtrrdi 2848	A membership and equality inference. (Contributed by NM, 24-Apr-2005.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 𝐶 = 𝐵    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	3eltr3i 2849	Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ 𝐴 ∈ 𝐵    &   ⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ 𝐶 ∈ 𝐷
Theorem	3eltr4i 2850	Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ 𝐴 ∈ 𝐵    &   ⊢ 𝐶 = 𝐴    &   ⊢ 𝐷 = 𝐵    ⇒   ⊢ 𝐶 ∈ 𝐷
Theorem	3eltr3d 2851	Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐶 ∈ 𝐷)
Theorem	3eltr4d 2852	Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐴)    &   ⊢ (𝜑 → 𝐷 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐶 ∈ 𝐷)
Theorem	3eltr3g 2853	Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) (Proof shortened by Wolf Lammen, 23-Nov-2019.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ (𝜑 → 𝐶 ∈ 𝐷)
Theorem	3eltr4g 2854	Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) (Proof shortened by Wolf Lammen, 23-Nov-2019.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 𝐶 = 𝐴    &   ⊢ 𝐷 = 𝐵    ⇒   ⊢ (𝜑 → 𝐶 ∈ 𝐷)
Theorem	eleq2s 2855	Substitution of equal classes into a membership antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝐴 ∈ 𝐵 → 𝜑)    &   ⊢ 𝐶 = 𝐵    ⇒   ⊢ (𝐴 ∈ 𝐶 → 𝜑)
Theorem	eqneltrd 2856	If a class is not an element of another class, an equal class is also not an element. Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐶)    ⇒   ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶)
Theorem	eqneltrrd 2857	If a class is not an element of another class, an equal class is also not an element. Deduction form. (Contributed by David Moews, 1-May-2017.) (Proof shortened by Wolf Lammen, 13-Nov-2019.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶)    ⇒   ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐶)
Theorem	neleqtrd 2858	If a class is not an element of another class, it is also not an element of an equal class. Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵)
Theorem	neleqtrrd 2859	If a class is not an element of another class, it is also not an element of an equal class. Deduction form. (Contributed by David Moews, 1-May-2017.) (Proof shortened by Wolf Lammen, 13-Nov-2019.)
⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)
Theorem	cleqh 2860*	Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions as in dfcleq 2729. See also cleqf 2935. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 14-Nov-2019.) Remove dependency on ax-13 2370. (Revised by BJ, 30-Nov-2020.)
⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)    &   ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵)    ⇒   ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Theorem	nelneq 2861	A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997.)
⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → ¬ 𝐴 = 𝐵)
Theorem	nelneq2 2862	A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)
⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → ¬ 𝐵 = 𝐶)
Theorem	eqsb1 2863*	Substitution for the left-hand side in an equality. Class version of equsb3 2100. (Contributed by Rodolfo Medina, 28-Apr-2010.)
⊢ ([𝑦 / 𝑥]𝑥 = 𝐴 ↔ 𝑦 = 𝐴)
Theorem	clelsb1 2864*	Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2113). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
Theorem	clelsb2 2865*	Substitution for the second argument of the membership predicate in an atomic formula (class version of elsb2 2122). (Contributed by Jim Kingdon, 22-Nov-2018.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 24-Nov-2024.)
⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦)
Theorem	clelsb2OLD 2866*	Obsolete version of clelsb2 2865 as of 24-Nov-2024.) (Contributed by Jim Kingdon, 22-Nov-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦)
Theorem	hbxfreq 2867	A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1826 for equivalence version. (Contributed by NM, 21-Aug-2007.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵)    ⇒   ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)
Theorem	hblem 2868*	Change the free variable of a hypothesis builder. (Contributed by NM, 21-Jun-1993.) (Revised by Andrew Salmon, 11-Jul-2011.) Add disjoint variable condition to avoid ax-13 2370. See hblemg 2869 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)    ⇒   ⊢ (𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴)
Theorem	hblemg 2869*	Change the free variable of a hypothesis builder. Usage of this theorem is discouraged because it depends on ax-13 2370. See hblem 2868 for a version with more disjoint variable conditions, but not requiring ax-13 2370. (Contributed by NM, 21-Jun-1993.) (Revised by Andrew Salmon, 11-Jul-2011.) (New usage is discouraged.)
⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)    ⇒   ⊢ (𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴)
2.1.2.4  Elementary properties of class abstractions
Theorem	abeq2 2870*	Equality of a class variable and a class abstraction (also called a class builder). Theorem 5.1 of [Quine] p. 34. This theorem shows the relationship between expressions with class abstractions and expressions with class variables. Note that abbi 2808 and its relatives are among those useful for converting theorems with class variables to equivalent theorems with wff variables, by first substituting a class abstraction for each class variable. Class variables can always be eliminated from a theorem to result in an equivalent theorem with wff variables, and vice-versa. The idea is roughly as follows. To convert a theorem with a wff variable 𝜑 (that has a free variable 𝑥) to a theorem with a class variable 𝐴, we substitute 𝑥 ∈ 𝐴 for 𝜑 throughout and simplify, where 𝐴 is a new class variable not already in the wff. An example is the conversion of zfauscl 5242 to inex1 5258 (look at the instance of zfauscl 5242 that occurs in the proof of inex1 5258). Conversely, to convert a theorem with a class variable 𝐴 to one with 𝜑, we substitute {𝑥 ∣ 𝜑} for 𝐴 throughout and simplify, where 𝑥 and 𝜑 are new setvar and wff variables not already in the wff. Examples include dfsymdif2 4196 and cp 9740; the latter derives a formula containing wff variables from substitution instances of the class variables in its equivalent formulation cplem2 9739. For more information on class variables, see Quine pp. 15-21 and/or Takeuti and Zaring pp. 10-13. Usage of abeq2w 2813 is preferred since it requires fewer axioms. (Contributed by NM, 26-May-1993.)
⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑))
Theorem	abeq1 2871*	Equality of a class variable and a class abstraction. Commuted form of abeq2 2870. (Contributed by NM, 20-Aug-1993.)
⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ ∀𝑥(𝜑 ↔ 𝑥 ∈ 𝐴))
Theorem	abeq2d 2872	Equality of a class variable and a class abstraction (deduction form of abeq2 2870). (Contributed by NM, 16-Nov-1995.)
⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓})    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓))
Theorem	abeq2i 2873	Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 3-Apr-1996.) (Proof shortened by Wolf Lammen, 15-Nov-2019.)
⊢ 𝐴 = {𝑥 ∣ 𝜑}    ⇒   ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)
Theorem	abeq1i 2874	Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 31-Jul-1994.) (Proof shortened by Wolf Lammen, 15-Nov-2019.)
⊢ {𝑥 ∣ 𝜑} = 𝐴    ⇒   ⊢ (𝜑 ↔ 𝑥 ∈ 𝐴)
Theorem	abbi2dv 2875*	Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) Avoid ax-11 2153. (Revised by Wolf Lammen, 6-May-2023.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓))    ⇒   ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓})
Theorem	abbi1dv 2876*	Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) (Proof shortened by Wolf Lammen, 16-Nov-2019.)
⊢ (𝜑 → (𝜓 ↔ 𝑥 ∈ 𝐴))    ⇒   ⊢ (𝜑 → {𝑥 ∣ 𝜓} = 𝐴)
Theorem	abbi2i 2877*	Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 26-May-1993.) Avoid ax-11 2153. (Revised by Wolf Lammen, 6-May-2023.)
⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)    ⇒   ⊢ 𝐴 = {𝑥 ∣ 𝜑}
Theorem	abid1 2878*	Every class is equal to a class abstraction (the class of sets belonging to it). Theorem 5.2 of [Quine] p. 35. This is a generalization to classes of cvjust 2730. The proof does not rely on cvjust 2730, so cvjust 2730 could be proved as a special instance of it. Note however that abid1 2878 necessarily relies on df-clel 2814, whereas cvjust 2730 does not. This theorem requires ax-ext 2707, df-clab 2714, df-cleq 2728, df-clel 2814, but to prove that any specific class term not containing class variables is a setvar or is equal to a class abstraction does not require these $a-statements. This last fact is a metatheorem, consequence of the fact that the only $a-statements with typecode class are cv 1539, cab 2713, and statements corresponding to defined class constructors. Note on the simultaneous presence in set.mm of this abid1 2878 and its commuted form abid2 2879: It is rare that two forms so closely related both appear in set.mm. Indeed, such equalities are generally used in later proofs as parts of transitive inferences, and with the many variants of eqtri 2764 (search for *eqtr*), it would be rare that either one would shorten a proof compared to the other. There is typically a choice between what we call a "definitional form", where the shorter expression is on the LHS (left-hand side), and a "computational form", where the shorter expression is on the RHS (right-hand side). An example is df-2 12129 versus 1p1e2 12191. We do not need 1p1e2 12191, but because it occurs "naturally" in computations, it can be useful to have it directly, together with a uniform set of 1-digit operations like 1p2e3 12209, etc. In most cases, we do not need both a definitional and a computational forms. A definitional form would favor consistency with genuine definitions, while a computational form is often more natural. The situation is similar with biconditionals in propositional calculus: see for instance pm4.24 564 and anidm 565, while other biconditionals generally appear in a single form (either definitional, but more often computational). In the present case, the equality is important enough that both abid1 2878 and abid2 2879 are in set.mm. (Contributed by NM, 26-Dec-1993.) (Revised by BJ, 10-Nov-2020.)
⊢ 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴}
Theorem	abid2 2879*	A simplification of class abstraction. Commuted form of abid1 2878. See comments there. (Contributed by NM, 26-Dec-1993.)
⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴
Theorem	clelab 2880*	Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) Avoid ax-11 2153, see sbc5ALT 3755 for more details. (Revised by SN, 2-Sep-2024.)
⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
Theorem	clelabOLD 2881*	Obsolete version of clelab 2880 as of 2-Sep-2024. (Contributed by NM, 23-Dec-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
Theorem	clabel 2882*	Membership of a class abstraction in another class. (Contributed by NM, 17-Jan-2006.)
⊢ ({𝑥 ∣ 𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑)))
Theorem	sbab 2883*	The right-hand side of the second equality is a way of representing proper substitution of 𝑦 for 𝑥 into a class variable. (Contributed by NM, 14-Sep-2003.)
⊢ (𝑥 = 𝑦 → 𝐴 = {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝐴})
2.1.3  Class form not-free predicate
Syntax	wnfc 2884	Extend wff definition to include the not-free predicate for classes.
wff Ⅎ𝑥𝐴
Theorem	nfcjust 2885*	Justification theorem for df-nfc 2886. (Contributed by Mario Carneiro, 13-Oct-2016.)
⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴)
Definition	df-nfc 2886*	Define the not-free predicate for classes. This is read "𝑥 is not free in 𝐴". Not-free means that the value of 𝑥 cannot affect the value of 𝐴, e.g., any occurrence of 𝑥 in 𝐴 is effectively bound by a "for all" or something that expands to one (such as "there exists"). It is defined in terms of the not-free predicate df-nf 1785 for wffs; see that definition for more information. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
Theorem	nfci 2887*	Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥 𝑦 ∈ 𝐴    ⇒   ⊢ Ⅎ𝑥𝐴
Theorem	nfcii 2888*	Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)    ⇒   ⊢ Ⅎ𝑥𝐴
Theorem	nfcr 2889*	Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) Drop ax-12 2170 but use ax-8 2107, df-clel 2814, and avoid a DV condition on 𝑦, 𝐴. (Revised by SN, 3-Jun-2024.)
⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴)
Theorem	nfcrALT 2890*	Alternate version of nfcr 2889. Avoids ax-8 2107 but uses ax-12 2170. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴)
Theorem	nfcri 2891*	Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-10 2136, ax-11 2153. (Revised by Gino Giotto, 23-May-2024.) Avoid ax-12 2170 (adopting Wolf Lammen's 13-May-2023 proof). (Revised by SN, 3-Jun-2024.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
Theorem	nfcd 2892*	Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥𝐴)
Theorem	nfcrd 2893*	Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)
Theorem	nfcriOLD 2894*	Obsolete version of nfcri 2891 as of 3-Jun-2024. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-10 2136, ax-11 2153. (Revised by Gino Giotto, 23-May-2024.) Avoid ax-12 2170. (Revised by SN, 26-May-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
Theorem	nfcriOLDOLD 2895*	Obsolete version of nfcri 2891 as of 26-May-2024. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-10 2136, ax-11 2153. (Revised by Gino Giotto, 23-May-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
Theorem	nfcrii 2896*	Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-10 2136, ax-11 2153. (Revised by Gino Giotto, 23-May-2024.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)
Theorem	nfcriiOLD 2897*	Obsolete version of nfcrii 2896 as of 23-May-2024. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)
Theorem	nfcriOLDOLDOLD 2898*	Obsolete version of nfcri 2891 as of 23-May-2024. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
Theorem	nfceqdf 2899	An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.) Avoid ax-8 2107 and df-clel 2814. (Revised by WL and SN, 23-Aug-2024.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵))
Theorem	nfceqdfOLD 2900	Obsolete version of nfceqdf 2899 as of 23-Aug-2024. (Contributed by Mario Carneiro, 14-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵))