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Theorem List for Metamath Proof Explorer - 2701-2800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem3eltr4i 2701 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝐴𝐵    &   𝐶 = 𝐴    &   𝐷 = 𝐵       𝐶𝐷

Theorem3eltr3d 2702 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐴𝐵)    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑𝐶𝐷)

Theorem3eltr4d 2703 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐴)    &   (𝜑𝐷 = 𝐵)       (𝜑𝐶𝐷)

Theorem3eltr3g 2704 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) (Proof shortened by Wolf Lammen, 23-Nov-2019.)
(𝜑𝐴𝐵)    &   𝐴 = 𝐶    &   𝐵 = 𝐷       (𝜑𝐶𝐷)

Theorem3eltr4g 2705 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) (Proof shortened by Wolf Lammen, 23-Nov-2019.)
(𝜑𝐴𝐵)    &   𝐶 = 𝐴    &   𝐷 = 𝐵       (𝜑𝐶𝐷)

Theoremeleq2s 2706 Substitution of equal classes into a membership antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝐴𝐵𝜑)    &   𝐶 = 𝐵       (𝐴𝐶𝜑)

Theoremeqneltrd 2707 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.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → ¬ 𝐵𝐶)       (𝜑 → ¬ 𝐴𝐶)

Theoremeqneltrrd 2708 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.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → ¬ 𝐴𝐶)       (𝜑 → ¬ 𝐵𝐶)

Theoremneleqtrd 2709 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 → ¬ 𝐶𝐴)    &   (𝜑𝐴 = 𝐵)       (𝜑 → ¬ 𝐶𝐵)

Theoremneleqtrrd 2710 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 → ¬ 𝐶𝐵)    &   (𝜑𝐴 = 𝐵)       (𝜑 → ¬ 𝐶𝐴)

Theoremcleqh 2711* Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2776. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 14-Nov-2019.) Remove dependency on ax-13 2234. (Revised by BJ, 30-Nov-2020.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)    &   (𝑦𝐵 → ∀𝑥 𝑦𝐵)       (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))

Theoremnelneq 2712 A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997.)
((𝐴𝐶 ∧ ¬ 𝐵𝐶) → ¬ 𝐴 = 𝐵)

Theoremnelneq2 2713 A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)
((𝐴𝐵 ∧ ¬ 𝐴𝐶) → ¬ 𝐵 = 𝐶)

Theoremeqsb3lem 2714* Lemma for eqsb3 2715. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
([𝑥 / 𝑦]𝑦 = 𝐴𝑥 = 𝐴)

Theoremeqsb3 2715* Substitution applied to an atomic wff (class version of equsb3 2420). (Contributed by Rodolfo Medina, 28-Apr-2010.)
([𝑥 / 𝑦]𝑦 = 𝐴𝑥 = 𝐴)

Theoremclelsb3 2716* Substitution applied to an atomic wff (class version of elsb3 2422). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
([𝑥 / 𝑦]𝑦𝐴𝑥𝐴)

Theoremhbxfreq 2717 A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1742 for equivalence version. (Contributed by NM, 21-Aug-2007.)
𝐴 = 𝐵    &   (𝑦𝐵 → ∀𝑥 𝑦𝐵)       (𝑦𝐴 → ∀𝑥 𝑦𝐴)

Theoremhblem 2718* Change the free variable of a hypothesis builder. Lemma for nfcrii 2744. (Contributed by NM, 21-Jun-1993.) (Revised by Andrew Salmon, 11-Jul-2011.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)       (𝑧𝐴 → ∀𝑥 𝑧𝐴)

Theoremabeq2 2719* 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 2724 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 𝑥𝐴 for 𝜑 throughout and simplify, where 𝐴 is a new class variable not already in the wff. An example is the conversion of zfauscl 4711 to inex1 4727 (look at the instance of zfauscl 4711 that occurs in the proof of inex1 4727). Conversely, to convert a theorem with a class variable 𝐴 to one with 𝜑, we substitute {𝑥𝜑} for 𝐴 throughout and simplify, where 𝑥 and 𝜑 are new setvar and wff variables not already in the wff. Examples include dfsymdif2 3813 and cp 8637; the latter derives a formula containing wff variables from substitution instances of the class variables in its equivalent formulation cplem2 8636. For more information on class variables, see Quine pp. 15-21 and/or Takeuti and Zaring pp. 10-13. (Contributed by NM, 26-May-1993.)

(𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))

Theoremabeq1 2720* Equality of a class variable and a class abstraction. Commuted form of abeq2 2719. (Contributed by NM, 20-Aug-1993.)
({𝑥𝜑} = 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))

Theoremabeq2d 2721 Equality of a class variable and a class abstraction (deduction form of abeq2 2719). (Contributed by NM, 16-Nov-1995.)
(𝜑𝐴 = {𝑥𝜓})       (𝜑 → (𝑥𝐴𝜓))

Theoremabeq2i 2722 Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 3-Apr-1996.) (Proof shortened by Wolf Lammen, 15-Nov-2019.)
𝐴 = {𝑥𝜑}       (𝑥𝐴𝜑)

Theoremabeq1i 2723 Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 31-Jul-1994.) (Proof shortened by Wolf Lammen, 15-Nov-2019.)
{𝑥𝜑} = 𝐴       (𝜑𝑥𝐴)

Theoremabbi 2724 Equivalent wff's correspond to equal class abstractions. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 16-Nov-2019.)
(∀𝑥(𝜑𝜓) ↔ {𝑥𝜑} = {𝑥𝜓})

Theoremabbi2i 2725* Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 26-May-1993.)
(𝑥𝐴𝜑)       𝐴 = {𝑥𝜑}

Theoremabbii 2726 Equivalent wff's yield equal class abstractions (inference rule). (Contributed by NM, 26-May-1993.)
(𝜑𝜓)       {𝑥𝜑} = {𝑥𝜓}

Theoremabbid 2727 Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 21-Jun-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝜓} = {𝑥𝜒})

Theoremabbidv 2728* Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 10-Aug-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝜓} = {𝑥𝜒})

Theoremabbi2dv 2729* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
(𝜑 → (𝑥𝐴𝜓))       (𝜑𝐴 = {𝑥𝜓})

Theoremabbi1dv 2730* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) (Proof shortened by Wolf Lammen, 16-Nov-2019.)
(𝜑 → (𝜓𝑥𝐴))       (𝜑 → {𝑥𝜓} = 𝐴)

Theoremabid1 2731* 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 2605. The proof does not rely on cvjust 2605, so cvjust 2605 could be proved as a special instance of it. Note however that abid1 2731 necessarily relies on df-clel 2606, whereas cvjust 2605 does not.

This theorem requires ax-ext 2590, df-clab 2597, df-cleq 2603, df-clel 2606, but to prove that any specific class term not containing class variables is a setvar or can be written as (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 1474, cab 2596 and statements corresponding to defined class constructors.

Note on the simultaneous presence in set.mm of this abid1 2731 and its commuted form abid2 2732: 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 2632 (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, and a "computational form" where the shorter expression is on the rhs. An example is df-2 10956 versus 1p1e2 11011. We do not need 1p1e2 11011, 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 11029, 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 computationa form is often more natural. The situation is similar with biconditionals in propositional calculus: see for instance pm4.24 673 and anidm 674, 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 2731 and abid2 2732 are in set.mm.

(Contributed by NM, 26-Dec-1993.) (Revised by BJ, 10-Nov-2020.)

𝐴 = {𝑥𝑥𝐴}

Theoremabid2 2732* A simplification of class abstraction. Commuted form of abid1 2731. See comments there. (Contributed by NM, 26-Dec-1993.)
{𝑥𝑥𝐴} = 𝐴

Theoremcbvab 2733 Rule used to change bound variables, using implicit substitution. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 16-Nov-2019.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝜑} = {𝑦𝜓}

Theoremcbvabv 2734* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-May-1999.)
(𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝜑} = {𝑦𝜓}

Theoremclelab 2735* Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2019.)
(𝐴 ∈ {𝑥𝜑} ↔ ∃𝑥(𝑥 = 𝐴𝜑))

Theoremclabel 2736* Membership of a class abstraction in another class. (Contributed by NM, 17-Jan-2006.)
({𝑥𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦𝐴 ∧ ∀𝑥(𝑥𝑦𝜑)))

Theoremsbab 2737* 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 = 𝑦𝐴 = {𝑧 ∣ [𝑦 / 𝑥]𝑧𝐴})

2.1.3  Class form not-free predicate

Syntaxwnfc 2738 Extend wff definition to include the not-free predicate for classes.
wff 𝑥𝐴

Theoremnfcjust 2739* Justification theorem for df-nfc 2740. (Contributed by Mario Carneiro, 13-Oct-2016.)
(∀𝑦𝑥 𝑦𝐴 ↔ ∀𝑧𝑥 𝑧𝐴)

Definitiondf-nfc 2740* 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 1701 for wffs; see that definition for more information. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)

Theoremnfci 2741* Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥 𝑦𝐴       𝑥𝐴

Theoremnfcii 2742* Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)       𝑥𝐴

Theoremnfcr 2743* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝑥𝐴 → Ⅎ𝑥 𝑦𝐴)

Theoremnfcrii 2744* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴       (𝑦𝐴 → ∀𝑥 𝑦𝐴)

Theoremnfcri 2745* Consequence of the not-free predicate. (Note that unlike nfcr 2743, this does not require 𝑦 and 𝐴 to be disjoint.) (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴       𝑥 𝑦𝐴

Theoremnfcd 2746* Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥 𝑦𝐴)       (𝜑𝑥𝐴)

Theoremnfceqdf 2747 An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑥𝜑    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝑥𝐴𝑥𝐵))

Theoremnfceqi 2748 Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 16-Nov-2019.)
𝐴 = 𝐵       (𝑥𝐴𝑥𝐵)

Theoremnfcxfr 2749 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝐴 = 𝐵    &   𝑥𝐵       𝑥𝐴

Theoremnfcxfrd 2750 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝐴 = 𝐵    &   (𝜑𝑥𝐵)       (𝜑𝑥𝐴)

Theoremnfcv 2751* If 𝑥 is disjoint from 𝐴, then 𝑥 is not free in 𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴

Theoremnfcvd 2752* If 𝑥 is disjoint from 𝐴, then 𝑥 is not free in 𝐴. (Contributed by Mario Carneiro, 7-Oct-2016.)
(𝜑𝑥𝐴)

Theoremnfab1 2753 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥{𝑥𝜑}

Theoremnfnfc1 2754 The setvar 𝑥 is bound in 𝑥𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝑥𝐴

Theoremnfab 2755 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑       𝑥{𝑦𝜑}

Theoremnfaba1 2756 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑥{𝑦 ∣ ∀𝑥𝜑}

Theoremnfcrd 2757* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝜑𝑥𝐴)       (𝜑 → Ⅎ𝑥 𝑦𝐴)

Theoremnfeqd 2758 Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑 → Ⅎ𝑥 𝐴 = 𝐵)

Theoremnfeld 2759 Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑 → Ⅎ𝑥 𝐴𝐵)

Theoremnfnfc 2760 Hypothesis builder for 𝑦𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-13 2234. (Revised by Wolf Lammen, 10-Dec-2019.)
𝑥𝐴       𝑥𝑦𝐴

TheoremnfnfcALT 2761 Alternate proof of nfnfc 2760. Shorter but requiring more axioms. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝐴       𝑥𝑦𝐴

Theoremnfeq 2762 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.)
𝑥𝐴    &   𝑥𝐵       𝑥 𝐴 = 𝐵

Theoremnfel 2763 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.)
𝑥𝐴    &   𝑥𝐵       𝑥 𝐴𝐵

Theoremnfeq1 2764* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
𝑥𝐴       𝑥 𝐴 = 𝐵

Theoremnfel1 2765* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
𝑥𝐴       𝑥 𝐴𝐵

Theoremnfeq2 2766* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
𝑥𝐵       𝑥 𝐴 = 𝐵

Theoremnfel2 2767* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
𝑥𝐵       𝑥 𝐴𝐵

Theoremdrnfc1 2768 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.)
(∀𝑥 𝑥 = 𝑦𝐴 = 𝐵)       (∀𝑥 𝑥 = 𝑦 → (𝑥𝐴𝑦𝐵))

Theoremdrnfc2 2769 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.)
(∀𝑥 𝑥 = 𝑦𝐴 = 𝐵)       (∀𝑥 𝑥 = 𝑦 → (𝑧𝐴𝑧𝐵))

Theoremnfabd2 2770 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)
𝑦𝜑    &   ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)       (𝜑𝑥{𝑦𝜓})

Theoremnfabd 2771 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑𝑥{𝑦𝜓})

Theoremdvelimdc 2772 Deduction form of dvelimc 2773. (Contributed by Mario Carneiro, 8-Oct-2016.)
𝑥𝜑    &   𝑧𝜑    &   (𝜑𝑥𝐴)    &   (𝜑𝑧𝐵)    &   (𝜑 → (𝑧 = 𝑦𝐴 = 𝐵))       (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐵))

Theoremdvelimc 2773 Version of dvelim 2325 for classes. (Contributed by Mario Carneiro, 8-Oct-2016.)
𝑥𝐴    &   𝑧𝐵    &   (𝑧 = 𝑦𝐴 = 𝐵)       (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐵)

Theoremnfcvf 2774 If 𝑥 and 𝑦 are distinct, then 𝑥 is not free in 𝑦. (Contributed by Mario Carneiro, 8-Oct-2016.)
(¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)

Theoremnfcvf2 2775 If 𝑥 and 𝑦 are distinct, then 𝑦 is not free in 𝑥. (Contributed by Mario Carneiro, 5-Dec-2016.)
(¬ ∀𝑥 𝑥 = 𝑦𝑦𝑥)

Theoremcleqf 2776 Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2711. (Contributed by NM, 26-May-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Nov-2019.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))

Theoremabid2f 2777 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, 17-Nov-2019.)
𝑥𝐴       {𝑥𝑥𝐴} = 𝐴

Theoremabeq2f 2778 Equality of a class variable and a class abstraction. In this version, the fact that 𝑥 is a non-free variable in 𝐴 is explicitely stated as a hypothesis. (Contributed by Thierry Arnoux, 11-May-2017.)
𝑥𝐴       (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))

Theoremsbabel 2779* 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, 26-Dec-2019.)
𝑥𝐴       ([𝑦 / 𝑥]{𝑧𝜑} ∈ 𝐴 ↔ {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴)

2.1.4  Negated equality and membership

Syntaxwne 2780 Extend wff notation to include inequality.
wff 𝐴𝐵

Syntaxwnel 2781 Extend wff notation to include negated membership.
wff 𝐴𝐵

Definitiondf-ne 2782 Define inequality. (Contributed by NM, 26-May-1993.)
(𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)

Definitiondf-nel 2783 Define negated membership. (Contributed by NM, 7-Aug-1994.)
(𝐴𝐵 ↔ ¬ 𝐴𝐵)

2.1.4.1  Negated equality

Theoremneii 2784 Inference associated with df-ne 2782. (Contributed by BJ, 7-Jul-2018.)
𝐴𝐵        ¬ 𝐴 = 𝐵

Theoremneir 2785 Inference associated with df-ne 2782. (Contributed by BJ, 7-Jul-2018.)
¬ 𝐴 = 𝐵       𝐴𝐵

Theoremnne 2786 Negation of inequality. (Contributed by NM, 9-Jun-2006.)
𝐴𝐵𝐴 = 𝐵)

Theoremneneqd 2787 Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝐴𝐵)       (𝜑 → ¬ 𝐴 = 𝐵)

Theoremneneq 2788 From inequality to non equality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴𝐵 → ¬ 𝐴 = 𝐵)

Theoremneqned 2789 If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2787. One-way deduction form of df-ne 2782. (Contributed by David Moews, 28-Feb-2017.) Allow a shortening of necon3bi 2808. (Revised by Wolf Lammen, 22-Nov-2019.)
(𝜑 → ¬ 𝐴 = 𝐵)       (𝜑𝐴𝐵)

Theoremneqne 2790 From non equality to inequality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐴 = 𝐵𝐴𝐵)

Theoremneirr 2791 No class is unequal to itself. Inequality is irreflexive. (Contributed by Stefan O'Rear, 1-Jan-2015.)
¬ 𝐴𝐴

Theoremexmidne 2792 Excluded middle with equality and inequality. (Contributed by NM, 3-Feb-2012.) (Proof shortened by Wolf Lammen, 17-Nov-2019.)
(𝐴 = 𝐵𝐴𝐵)

Theoremeqneqall 2793 A contradiction concerning equality implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
(𝐴 = 𝐵 → (𝐴𝐵𝜑))

Theoremnonconne 2794 Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.) (Proof shortened by Wolf Lammen, 21-Dec-2019.)
¬ (𝐴 = 𝐵𝐴𝐵)

Theoremnecon3ad 2795 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 → (𝜓𝐴 = 𝐵))       (𝜑 → (𝐴𝐵 → ¬ 𝜓))

Theoremnecon3bd 2796 Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑 → (𝐴 = 𝐵𝜓))       (𝜑 → (¬ 𝜓𝐴𝐵))

Theoremnecon2ad 2797 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 → (𝜓𝐴𝐵))

Theoremnecon2bd 2798 Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)
(𝜑 → (𝜓𝐴𝐵))       (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓))

Theoremnecon1ad 2799 Contrapositive deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Wolf Lammen, 23-Nov-2019.)
(𝜑 → (¬ 𝜓𝐴 = 𝐵))       (𝜑 → (𝐴𝐵𝜓))

Theoremnecon1bd 2800 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 → (𝐴𝐵𝜓))       (𝜑 → (¬ 𝜓𝐴 = 𝐵))

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