P. 30 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 2901-3000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	eqabb 2901*	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 abbib 2831 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 sepgi 5259 to inex1 5285 (look at the instance of sepgi 5259 that occurs in the proof of inex1 5285). 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 4213 and cp 9881; the latter derives a formula containing wff variables from substitution instances of the class variables in its equivalent formulation cplem2 9879. For more information on class variables, see Quine pp. 15-21 and/or Takeuti and Zaring pp. 10-13. Usage of eqabbw 2835 is preferred since it requires fewer axioms. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 12-Feb-2025.)
⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑))
Theorem	eqabcb 2902*	Equality of a class variable and a class abstraction. Commuted form of eqabb 2901. (Contributed by NM, 20-Aug-1993.)
⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ ∀𝑥(𝜑 ↔ 𝑥 ∈ 𝐴))
Theorem	eqabrd 2903	Equality of a class variable and a class abstraction (deduction form of eqabb 2901). (Contributed by NM, 16-Nov-1995.)
⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓})    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓))
Theorem	eqabri 2904	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	eqabcri 2905	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	clelab 2906*	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 2191, see sbc5ALT 3772 for more details. (Revised by SN, 2-Sep-2024.)
⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
Theorem	clabel 2907*	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 2908*	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 2909	Extend wff definition to include the not-free predicate for classes.
wff Ⅎ𝑥𝐴
Theorem	nfcjust 2910*	Justification theorem for df-nfc 2911. (Contributed by Mario Carneiro, 13-Oct-2016.)
⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴)
Definition	df-nfc 2911*	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 1813 for wffs; see that definition for more information. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
Theorem	nfci 2912*	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 2913*	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 2914*	Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) Drop ax-12 2212 but use ax-8 2144, df-clel 2837, and avoid a DV condition on 𝑦, 𝐴. (Revised by SN, 3-Jun-2024.)
⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴)
Theorem	nfcrALT 2915*	Alternate version of nfcr 2914. Avoids ax-8 2144 but uses ax-12 2212. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴)
Theorem	nfcri 2916*	Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-10 2175, ax-11 2191. (Revised by GG, 23-May-2024.) Avoid ax-12 2212 (adopting Wolf Lammen's 13-May-2023 proof). (Revised by SN, 3-Jun-2024.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
Theorem	nfcd 2917*	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 2918*	Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)
Theorem	nfcrii 2919*	Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-10 2175, ax-11 2191. (Revised by GG, 23-May-2024.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)
Theorem	nfceqdf 2920	An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.) Avoid ax-8 2144 and df-clel 2837. (Revised by WL and SN, 23-Aug-2024.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵))
Theorem	nfceqi 2921	Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) Avoid ax-12 2212. (Revised by Wolf Lammen, 19-Jun-2023.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)
Theorem	nfcxfr 2922	A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ 𝐴 = 𝐵    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥𝐴
Theorem	nfcxfrd 2923	A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥𝐴)
Theorem	nfcv 2924*	If 𝑥 is disjoint from 𝐴, then 𝑥 is not free in 𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥𝐴
Theorem	nfcvd 2925*	If 𝑥 is disjoint from 𝐴, then 𝑥 is not free in 𝐴. (Contributed by Mario Carneiro, 7-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐴)
Theorem	nfab1 2926	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
Theorem	nfnfc1 2927	The setvar 𝑥 is bound in Ⅎ𝑥𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥Ⅎ𝑥𝐴
Theorem	clelsb1fw 2928*	Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2150). Version of clelsb1f 2929 with a disjoint variable condition, which does not require ax-13 2403. (Contributed by Rodolfo Medina, 28-Apr-2010.) Avoid ax-13 2403. (Revised by GG, 10-Jan-2024.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
Theorem	clelsb1f 2929	Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2150). Usage of this theorem is discouraged because it depends on ax-13 2403. See clelsb1fw 2928 not requiring ax-13 2403, but extra disjoint variables. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.) (Proof shortened by Wolf Lammen, 7-May-2023.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
Theorem	nfab 2930*	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Add disjoint variable condition to avoid ax-13 2403. See nfabg 2931 for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥{𝑦 ∣ 𝜑}
Theorem	nfabg 2931	Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2403. See nfab 2930 for a version with more disjoint variable conditions, but not requiring ax-13 2403. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥{𝑦 ∣ 𝜑}
Theorem	nfaba1 2932*	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2403. See nfaba1g 2933 for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024.) Avoid ax-12 2212. (Revised by SN, 14-May-2025.)
⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥𝜑}
Theorem	nfaba1g 2933	Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2403. See nfaba1 2932 for a version with a disjoint variable condition, but not requiring ax-13 2403. (Contributed by Mario Carneiro, 14-Oct-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥𝜑}
Theorem	nfeqd 2934	Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥 𝐴 = 𝐵)
Theorem	nfeld 2935	Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ 𝐵)
Theorem	nfnfc 2936	Hypothesis builder for Ⅎ𝑦𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-13 2403. (Revised by Wolf Lammen, 10-Dec-2019.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥Ⅎ𝑦𝐴
Theorem	nfeq 2937	Hypothesis builder for equality. (Contributed by NM, 21-Jun-1993.) (Revised by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 16-Nov-2019.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 𝐴 = 𝐵
Theorem	nfel 2938	Hypothesis builder for elementhood. (Contributed by NM, 1-Aug-1993.) (Revised by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 16-Nov-2019.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
Theorem	nfeq1 2939*	Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥 𝐴 = 𝐵
Theorem	nfel1 2940*	Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
Theorem	nfeq2 2941*	Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 𝐴 = 𝐵
Theorem	nfel2 2942*	Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
Theorem	drnfc1 2943	Formula-building lemma for use with the Distinctor Reduction Theorem. Usage of this theorem is discouraged because it depends on ax-13 2403. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-8 2144, ax-11 2191. (Revised by Wolf Lammen, 22-Sep-2024.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑦𝐵))
Theorem	drnfc2 2944	Formula-building lemma for use with the Distinctor Reduction Theorem. Proof revision is marked as discouraged because the minimizer replaces albidv 1949 with dral2 2469, leading to a one byte longer proof. However feel free to manually edit it according to conventions. (TODO: dral2 2469 depends on ax-13 2403, hence its usage during minimizing is discouraged. Check in the long run whether this is a permanent restriction). Usage of this theorem is discouraged because it depends on ax-13 2403. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-8 2144. (Revised by Wolf Lammen, 22-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝐴 ↔ Ⅎ𝑧𝐵))
Theorem	nfabdw 2945*	Bound-variable hypothesis builder for a class abstraction. Version of nfabd 2946 with a disjoint variable condition, which does not require ax-13 2403. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-13 2403. (Revised by GG, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 23-Sep-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})
Theorem	nfabd 2946	Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2403. Use the weaker nfabdw 2945 when possible. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-9 2152 and ax-ext 2734. (Revised by Wolf Lammen, 23-May-2023.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})
Theorem	nfabd2 2947	Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2403. (Contributed by Mario Carneiro, 8-Oct-2016.) (Proof shortened by Wolf Lammen, 10-May-2023.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})
Theorem	dvelimdc 2948	Deduction form of dvelimc 2949. Usage of this theorem is discouraged because it depends on ax-13 2403. (Contributed by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑧𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑧𝐵)    &   ⊢ (𝜑 → (𝑧 = 𝑦 → 𝐴 = 𝐵))    ⇒   ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵))
Theorem	dvelimc 2949	Version of dvelim 2482 for classes. Usage of this theorem is discouraged because it depends on ax-13 2403. (Contributed by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑧𝐵    &   ⊢ (𝑧 = 𝑦 → 𝐴 = 𝐵)    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵)
Theorem	nfcvf 2950	If 𝑥 and 𝑦 are distinct, then 𝑥 is not free in 𝑦. Usage of this theorem is discouraged because it depends on ax-13 2403. See nfcv 2924 for a version that replaces the distinctor with a disjoint variable condition, requiring fewer axioms. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-ext 2734. (Revised by Wolf Lammen, 10-May-2023.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
Theorem	nfcvf2 2951	If 𝑥 and 𝑦 are distinct, then 𝑦 is not free in 𝑥. Usage of this theorem is discouraged because it depends on ax-13 2403. See nfcv 2924 for a version that replaces the distinctor with a disjoint variable condition, requiring fewer axioms. (Contributed by Mario Carneiro, 5-Dec-2016.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥)
Theorem	cleqf 2952	Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions as in dfcleq 2755. See also cleqh 2891. (Contributed by NM, 26-May-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Nov-2019.) Avoid ax-13 2403. (Revised by Wolf Lammen, 10-May-2023.) Avoid ax-10 2175. (Revised by GG, 20-Aug-2023.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Theorem	eqabf 2953	Equality of a class variable and a class abstraction. In this version, the fact that 𝑥 is a nonfree variable in 𝐴 is explicitly stated as a hypothesis. (Contributed by Thierry Arnoux, 11-May-2017.) Avoid ax-13 2403. (Revised by Wolf Lammen, 13-May-2023.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑))
Theorem	abid2f 2954	A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 5-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 26-Feb-2025.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴
Theorem	abid2fOLD 2955	Obsolete version of abid2f 2954 as of 26-Feb-2025. (Contributed by NM, 5-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴
Theorem	sbabel 2956*	Theorem to move a substitution in and out of a class abstraction. (Contributed by NM, 27-Sep-2003.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 28-Oct-2024.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ ([𝑦 / 𝑥]{𝑧 ∣ 𝜑} ∈ 𝐴 ↔ {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴)
2.1.4  Negated equality and membership
2.1.4.1  Negated equality
Syntax	wne 2957	Extend wff notation to include inequality.
wff 𝐴 ≠ 𝐵
Definition	df-ne 2958	Define inequality. (Contributed by NM, 26-May-1993.)
⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
Theorem	neii 2959	Inference associated with df-ne 2958. (Contributed by BJ, 7-Jul-2018.)
⊢ 𝐴 ≠ 𝐵    ⇒   ⊢ ¬ 𝐴 = 𝐵
Theorem	neir 2960	Inference associated with df-ne 2958. (Contributed by BJ, 7-Jul-2018.)
⊢ ¬ 𝐴 = 𝐵    ⇒   ⊢ 𝐴 ≠ 𝐵
Theorem	nne 2961	Negation of inequality. (Contributed by NM, 9-Jun-2006.)
⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵)
Theorem	neneqd 2962	Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
Theorem	neneq 2963	From inequality to non-equality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 = 𝐵)
Theorem	neqned 2964	If it is not the case that two classes are equal, then they are unequal. Converse of neneqd 2962. One-way deduction form of df-ne 2958. (Contributed by David Moews, 28-Feb-2017.) Allow a shortening of necon3bi 2983. (Revised by Wolf Lammen, 22-Nov-2019.)
⊢ (𝜑 → ¬ 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐵)
Theorem	neqne 2965	From non-equality to inequality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (¬ 𝐴 = 𝐵 → 𝐴 ≠ 𝐵)
Theorem	neirr 2966	No class is unequal to itself. Inequality is irreflexive. (Contributed by Stefan O'Rear, 1-Jan-2015.)
⊢ ¬ 𝐴 ≠ 𝐴
Theorem	exmidne 2967	Excluded middle with equality and inequality. (Contributed by NM, 3-Feb-2012.) (Proof shortened by Wolf Lammen, 17-Nov-2019.)
⊢ (𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵)
Theorem	eqneqall 2968	A contradiction concerning equality implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝜑))
Theorem	nonconne 2969	Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.) (Proof shortened by Wolf Lammen, 21-Dec-2019.)
⊢ ¬ (𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐵)
Theorem	necon3ad 2970	Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 23-Nov-2019.)
⊢ (𝜑 → (𝜓 → 𝐴 = 𝐵))    ⇒   ⊢ (𝜑 → (𝐴 ≠ 𝐵 → ¬ 𝜓))
Theorem	necon3bd 2971	Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (𝜑 → (𝐴 = 𝐵 → 𝜓))    ⇒   ⊢ (𝜑 → (¬ 𝜓 → 𝐴 ≠ 𝐵))
Theorem	necon2ad 2972	Contrapositive inference for inequality. (Contributed by NM, 19-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 23-Nov-2019.)
⊢ (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓))    ⇒   ⊢ (𝜑 → (𝜓 → 𝐴 ≠ 𝐵))
Theorem	necon2bd 2973	Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)
⊢ (𝜑 → (𝜓 → 𝐴 ≠ 𝐵))    ⇒   ⊢ (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓))
Theorem	necon1ad 2974	Contrapositive deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Wolf Lammen, 23-Nov-2019.)
⊢ (𝜑 → (¬ 𝜓 → 𝐴 = 𝐵))    ⇒   ⊢ (𝜑 → (𝐴 ≠ 𝐵 → 𝜓))
Theorem	necon1bd 2975	Contrapositive deduction for inequality. (Contributed by NM, 21-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 23-Nov-2019.)
⊢ (𝜑 → (𝐴 ≠ 𝐵 → 𝜓))    ⇒   ⊢ (𝜑 → (¬ 𝜓 → 𝐴 = 𝐵))
Theorem	necon4ad 2976	Contrapositive inference for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 23-Nov-2019.)
⊢ (𝜑 → (𝐴 ≠ 𝐵 → ¬ 𝜓))    ⇒   ⊢ (𝜑 → (𝜓 → 𝐴 = 𝐵))
Theorem	necon4bd 2977	Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 23-Nov-2019.)
⊢ (𝜑 → (¬ 𝜓 → 𝐴 ≠ 𝐵))    ⇒   ⊢ (𝜑 → (𝐴 = 𝐵 → 𝜓))
Theorem	necon3d 2978	Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.)
⊢ (𝜑 → (𝐴 = 𝐵 → 𝐶 = 𝐷))    ⇒   ⊢ (𝜑 → (𝐶 ≠ 𝐷 → 𝐴 ≠ 𝐵))
Theorem	necon1d 2979	Contrapositive law deduction for inequality. (Contributed by NM, 28-Dec-2008.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (𝜑 → (𝐴 ≠ 𝐵 → 𝐶 = 𝐷))    ⇒   ⊢ (𝜑 → (𝐶 ≠ 𝐷 → 𝐴 = 𝐵))
Theorem	necon2d 2980	Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.)
⊢ (𝜑 → (𝐴 = 𝐵 → 𝐶 ≠ 𝐷))    ⇒   ⊢ (𝜑 → (𝐶 = 𝐷 → 𝐴 ≠ 𝐵))
Theorem	necon4d 2981	Contrapositive inference for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (𝜑 → (𝐴 ≠ 𝐵 → 𝐶 ≠ 𝐷))    ⇒   ⊢ (𝜑 → (𝐶 = 𝐷 → 𝐴 = 𝐵))
Theorem	necon3ai 2982	Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 28-Oct-2024.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝐴 ≠ 𝐵 → ¬ 𝜑)
Theorem	necon3bi 2983	Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 22-Nov-2019.)
⊢ (𝐴 = 𝐵 → 𝜑)    ⇒   ⊢ (¬ 𝜑 → 𝐴 ≠ 𝐵)
Theorem	necon1ai 2984	Contrapositive inference for inequality. (Contributed by NM, 12-Feb-2007.) (Proof shortened by Wolf Lammen, 22-Nov-2019.)
⊢ (¬ 𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝐴 ≠ 𝐵 → 𝜑)
Theorem	necon1bi 2985	Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 22-Nov-2019.)
⊢ (𝐴 ≠ 𝐵 → 𝜑)    ⇒   ⊢ (¬ 𝜑 → 𝐴 = 𝐵)
Theorem	necon2ai 2986	Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 22-Nov-2019.)
⊢ (𝐴 = 𝐵 → ¬ 𝜑)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐵)
Theorem	necon2bi 2987	Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    ⇒   ⊢ (𝐴 = 𝐵 → ¬ 𝜑)
Theorem	necon4ai 2988	Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 22-Nov-2019.)
⊢ (𝐴 ≠ 𝐵 → ¬ 𝜑)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	necon3i 2989	Contrapositive inference for inequality. (Contributed by NM, 9-Aug-2006.) (Proof shortened by Wolf Lammen, 22-Nov-2019.)
⊢ (𝐴 = 𝐵 → 𝐶 = 𝐷)    ⇒   ⊢ (𝐶 ≠ 𝐷 → 𝐴 ≠ 𝐵)
Theorem	necon1i 2990	Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)
⊢ (𝐴 ≠ 𝐵 → 𝐶 = 𝐷)    ⇒   ⊢ (𝐶 ≠ 𝐷 → 𝐴 = 𝐵)
Theorem	necon2i 2991	Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)
⊢ (𝐴 = 𝐵 → 𝐶 ≠ 𝐷)    ⇒   ⊢ (𝐶 = 𝐷 → 𝐴 ≠ 𝐵)
Theorem	necon4i 2992	Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)
⊢ (𝐴 ≠ 𝐵 → 𝐶 ≠ 𝐷)    ⇒   ⊢ (𝐶 = 𝐷 → 𝐴 = 𝐵)
Theorem	necon3abid 2993	Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.)
⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝜓))    ⇒   ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ ¬ 𝜓))
Theorem	necon3bbid 2994	Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007.)
⊢ (𝜑 → (𝜓 ↔ 𝐴 = 𝐵))    ⇒   ⊢ (𝜑 → (¬ 𝜓 ↔ 𝐴 ≠ 𝐵))
Theorem	necon1abid 2995	Contrapositive deduction for inequality. (Contributed by NM, 21-Aug-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)
⊢ (𝜑 → (¬ 𝜓 ↔ 𝐴 = 𝐵))    ⇒   ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝜓))
Theorem	necon1bbid 2996	Contrapositive inference for inequality. (Contributed by NM, 31-Jan-2008.)
⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝜓))    ⇒   ⊢ (𝜑 → (¬ 𝜓 ↔ 𝐴 = 𝐵))
Theorem	necon4abid 2997	Contrapositive law deduction for inequality. (Contributed by NM, 11-Jan-2008.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)
⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ ¬ 𝜓))    ⇒   ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝜓))
Theorem	necon4bbid 2998	Contrapositive law deduction for inequality. (Contributed by NM, 9-May-2012.)
⊢ (𝜑 → (¬ 𝜓 ↔ 𝐴 ≠ 𝐵))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝐴 = 𝐵))
Theorem	necon2abid 2999	Contrapositive deduction for inequality. (Contributed by NM, 18-Jul-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)
⊢ (𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝜓))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝐴 ≠ 𝐵))
Theorem	necon2bbid 3000	Contrapositive deduction for inequality. (Contributed by NM, 13-Apr-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)
⊢ (𝜑 → (𝜓 ↔ 𝐴 ≠ 𝐵))    ⇒   ⊢ (𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝜓))