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

Theoremeqeltrdi 2901 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
(𝜑𝐴 = 𝐵)    &   𝐵𝐶       (𝜑𝐴𝐶)

Theoremeqeltrrdi 2902 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
(𝜑𝐵 = 𝐴)    &   𝐵𝐶       (𝜑𝐴𝐶)

Theoremeleqtrdi 2903 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
(𝜑𝐴𝐵)    &   𝐵 = 𝐶       (𝜑𝐴𝐶)

Theoremeleqtrrdi 2904 A membership and equality inference. (Contributed by NM, 24-Apr-2005.)
(𝜑𝐴𝐵)    &   𝐶 = 𝐵       (𝜑𝐴𝐶)

Theorem3eltr3i 2905 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝐴𝐵    &   𝐴 = 𝐶    &   𝐵 = 𝐷       𝐶𝐷

Theorem3eltr4i 2906 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝐴𝐵    &   𝐶 = 𝐴    &   𝐷 = 𝐵       𝐶𝐷

Theorem3eltr3d 2907 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐴𝐵)    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑𝐶𝐷)

Theorem3eltr4d 2908 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐴)    &   (𝜑𝐷 = 𝐵)       (𝜑𝐶𝐷)

Theorem3eltr3g 2909 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 2910 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 2911 Substitution of equal classes into a membership antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝐴𝐵𝜑)    &   𝐶 = 𝐵       (𝐴𝐶𝜑)

Theoremeqneltrd 2912 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 2913 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 2914 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 2915 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 2916* Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions as in dfcleq 2795. See also cleqf 2986. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 14-Nov-2019.) Remove dependency on ax-13 2382. (Revised by BJ, 30-Nov-2020.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)    &   (𝑦𝐵 → ∀𝑥 𝑦𝐵)       (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))

Theoremnelneq 2917 A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997.)
((𝐴𝐶 ∧ ¬ 𝐵𝐶) → ¬ 𝐴 = 𝐵)

Theoremnelneq2 2918 A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)
((𝐴𝐵 ∧ ¬ 𝐴𝐶) → ¬ 𝐵 = 𝐶)

Theoremeqsb3 2919* Substitution applied to an atomic wff (class version of equsb3 2107). (Contributed by Rodolfo Medina, 28-Apr-2010.)
([𝑦 / 𝑥]𝑥 = 𝐴𝑦 = 𝐴)

Theoremclelsb3 2920* Substitution applied to an atomic wff (class version of elsb3 2120). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)

Theoremclelsb3vOLD 2921* Obsolete version of clelsb3 2920 as of 29-Jul-2023. (Contributed by Wolf Lammen, 30-Apr-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)

Theoremhbxfreq 2922 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    &   (𝑦𝐵 → ∀𝑥 𝑦𝐵)       (𝑦𝐴 → ∀𝑥 𝑦𝐴)

Theoremhblem 2923* 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 2382. See hblemg 2924 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)       (𝑧𝐴 → ∀𝑥 𝑧𝐴)

Theoremhblemg 2924* Change the free variable of a hypothesis builder. Usage of this theorem is discouraged because it depends on ax-13 2382. See hblem 2923 for a version with more disjoint variable conditions, but not requiring ax-13 2382. (Contributed by NM, 21-Jun-1993.) (Revised by Andrew Salmon, 11-Jul-2011.) (New usage is discouraged.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)       (𝑧𝐴 → ∀𝑥 𝑧𝐴)

2.1.2.4  Elementary properties of class abstractions

Theoremabeq2 2925* 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 2868 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 5172 to inex1 5188 (look at the instance of zfauscl 5172 that occurs in the proof of inex1 5188). 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 4180 and cp 9308; the latter derives a formula containing wff variables from substitution instances of the class variables in its equivalent formulation cplem2 9307. 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 2926* Equality of a class variable and a class abstraction. Commuted form of abeq2 2925. (Contributed by NM, 20-Aug-1993.)
({𝑥𝜑} = 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))

Theoremabeq2d 2927 Equality of a class variable and a class abstraction (deduction form of abeq2 2925). (Contributed by NM, 16-Nov-1995.)
(𝜑𝐴 = {𝑥𝜓})       (𝜑 → (𝑥𝐴𝜓))

Theoremabeq2i 2928 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 = {𝑥𝜑}       (𝑥𝐴𝜑)

Theoremabeq1i 2929 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.)
{𝑥𝜑} = 𝐴       (𝜑𝑥𝐴)

Theoremabbi2dv 2930* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) Avoid ax-11 2159. (Revised by Wolf Lammen, 6-May-2023.)
(𝜑 → (𝑥𝐴𝜓))       (𝜑𝐴 = {𝑥𝜓})

Theoremabbi1dv 2931* 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)

Theoremabbi2i 2932* Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 26-May-1993.) Avoid ax-11 2159. (Revised by Wolf Lammen, 6-May-2023.)
(𝑥𝐴𝜑)       𝐴 = {𝑥𝜑}

TheoremabbiOLD 2933 Obsolete proof of abbi 2868 as of 7-Jan-2024. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥(𝜑𝜓) ↔ {𝑥𝜑} = {𝑥𝜓})

Theoremabid1 2934* 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 2796. The proof does not rely on cvjust 2796, so cvjust 2796 could be proved as a special instance of it. Note however that abid1 2934 necessarily relies on df-clel 2873, whereas cvjust 2796 does not.

This theorem requires ax-ext 2773, df-clab 2780, df-cleq 2794, df-clel 2873, 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 1537, cab 2779, and statements corresponding to defined class constructors.

Note on the simultaneous presence in set.mm of this abid1 2934 and its commuted form abid2 2935: 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 2824 (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 11692 versus 1p1e2 11754. We do not need 1p1e2 11754, 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 11772, 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 pm4.24 567 and anidm 568, 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 2934 and abid2 2935 are in set.mm.

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

𝐴 = {𝑥𝑥𝐴}

Theoremabid2 2935* A simplification of class abstraction. Commuted form of abid1 2934. See comments there. (Contributed by NM, 26-Dec-1993.)
{𝑥𝑥𝐴} = 𝐴

Theoremclelab 2936* 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 2937* Membership of a class abstraction in another class. (Contributed by NM, 17-Jan-2006.)
({𝑥𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦𝐴 ∧ ∀𝑥(𝑥𝑦𝜑)))

Theoremsbab 2938* 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 2939 Extend wff definition to include the not-free predicate for classes.
wff 𝑥𝐴

Theoremnfcjust 2940* Justification theorem for df-nfc 2941. (Contributed by Mario Carneiro, 13-Oct-2016.)
(∀𝑦𝑥 𝑦𝐴 ↔ ∀𝑧𝑥 𝑧𝐴)

Definitiondf-nfc 2941* 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 1786 for wffs; see that definition for more information. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)

Theoremnfci 2942* Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥 𝑦𝐴       𝑥𝐴

Theoremnfcii 2943* Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)       𝑥𝐴

Theoremnfcr 2944* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) Drop ax-12 2176 but use ax-8 2114, and avoid a DV condition on 𝑦, 𝐴. (Revised by SN, 3-Jun-2024.)
(𝑥𝐴 → Ⅎ𝑥 𝑦𝐴)

TheoremnfcrALT 2945* Alternate version of nfcr 2944. Avoids ax-8 2114 but uses ax-12 2176. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝑥𝐴 → Ⅎ𝑥 𝑦𝐴)

Theoremnfcri 2946* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-10 2143, ax-11 2159. (Revised by Gino Giotto, 23-May-2024.) Avoid ax-12 2176 (adopting Wolf Lammen's 13-May-2023 proof). (Revised by SN, 3-Jun-2024.)
𝑥𝐴       𝑥 𝑦𝐴

Theoremnfcd 2947* Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥 𝑦𝐴)       (𝜑𝑥𝐴)

Theoremnfcrd 2948* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝜑𝑥𝐴)       (𝜑 → Ⅎ𝑥 𝑦𝐴)

TheoremnfcriOLD 2949* Obsolete version of nfcri 2946 as of 3-Jun-2024. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-10 2143, ax-11 2159. (Revised by Gino Giotto, 23-May-2024.) Avoid ax-12 2176. (Revised by SN, 26-May-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑥𝐴       𝑥 𝑦𝐴

TheoremnfcriOLDOLD 2950* Obsolete version of nfcri 2946 as of 26-May-2024. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-10 2143, ax-11 2159. (Revised by Gino Giotto, 23-May-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑥𝐴       𝑥 𝑦𝐴

Theoremnfcrii 2951* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-10 2143, ax-11 2159. (Revised by Gino Giotto, 23-May-2024.)
𝑥𝐴       (𝑦𝐴 → ∀𝑥 𝑦𝐴)

TheoremnfcriiOLD 2952* Obsolete version of nfcrii 2951 as of 23-May-2024. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝐴       (𝑦𝐴 → ∀𝑥 𝑦𝐴)

TheoremnfcriOLDOLDOLD 2953* Obsolete version of nfcri 2946 as of 23-May-2024. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝐴       𝑥 𝑦𝐴

Theoremnfceqdf 2954 An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑥𝜑    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝑥𝐴𝑥𝐵))

Theoremnfceqi 2955 Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) Avoid ax-12 2176. (Revised by Wolf Lammen, 19-Jun-2023.)
𝐴 = 𝐵       (𝑥𝐴𝑥𝐵)

Theoremnfcxfr 2956 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 2957 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 2958* If 𝑥 is disjoint from 𝐴, then 𝑥 is not free in 𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴

Theoremnfcvd 2959* If 𝑥 is disjoint from 𝐴, then 𝑥 is not free in 𝐴. (Contributed by Mario Carneiro, 7-Oct-2016.)
(𝜑𝑥𝐴)

Theoremnfab1 2960 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥{𝑥𝜑}

Theoremnfnfc1 2961 The setvar 𝑥 is bound in 𝑥𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝑥𝐴

Theoremclelsb3fw 2962* Substitution applied to an atomic wff (class version of elsb3 2120). Version of clelsb3f 2963 with a disjoint variable condition, which does not require ax-13 2382. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Revised by Gino Giotto, 10-Jan-2024.)
𝑥𝐴       ([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)

Theoremclelsb3f 2963 Substitution applied to an atomic wff (class version of elsb3 2120). Usage of this theorem is discouraged because it depends on ax-13 2382. See clelsb3fw 2962 not requiring ax-13 2382, 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.)
𝑥𝐴       ([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)

Theoremnfab 2964* Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Add disjoint variable condition to avoid ax-13 2382. See nfabg 2965 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
𝑥𝜑       𝑥{𝑦𝜑}

Theoremnfabg 2965 Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2382. See nfab 2964 for a version with more disjoint variable conditions, but not requiring ax-13 2382. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.)
𝑥𝜑       𝑥{𝑦𝜑}

Theoremnfaba1 2966* Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2382. See nfaba1g 2967 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
𝑥{𝑦 ∣ ∀𝑥𝜑}

Theoremnfaba1g 2967 Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2382. See nfaba1 2966 for a version with a disjoint variable condition, but not requiring ax-13 2382. (Contributed by Mario Carneiro, 14-Oct-2016.) (New usage is discouraged.)
𝑥{𝑦 ∣ ∀𝑥𝜑}

Theoremnfeqd 2968 Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑 → Ⅎ𝑥 𝐴 = 𝐵)

Theoremnfeld 2969 Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑 → Ⅎ𝑥 𝐴𝐵)

Theoremnfnfc 2970 Hypothesis builder for 𝑦𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-13 2382. (Revised by Wolf Lammen, 10-Dec-2019.)
𝑥𝐴       𝑥𝑦𝐴

Theoremnfeq 2971 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 2972 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 2973* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
𝑥𝐴       𝑥 𝐴 = 𝐵

Theoremnfel1 2974* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
𝑥𝐴       𝑥 𝐴𝐵

Theoremnfeq2 2975* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
𝑥𝐵       𝑥 𝐴 = 𝐵

Theoremnfel2 2976* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
𝑥𝐵       𝑥 𝐴𝐵

Theoremdrnfc1 2977 Formula-building lemma for use with the Distinctor Reduction Theorem. Usage of this theorem is discouraged because it depends on ax-13 2382. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-11 2159. (Revised by Wolf Lammen, 10-May-2023.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦𝐴 = 𝐵)       (∀𝑥 𝑥 = 𝑦 → (𝑥𝐴𝑦𝐵))

Theoremdrnfc2 2978 Formula-building lemma for use with the Distinctor Reduction Theorem. Proof revision is marked as discouraged because the minimizer replaces albidv 1921 with dral2 2452, leading to a one byte longer proof. However feel free to manually edit it according to conventions. Usage of this theorem is discouraged because it depends on ax-13 2382. (Contributed by Mario Carneiro, 8-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦𝐴 = 𝐵)       (∀𝑥 𝑥 = 𝑦 → (𝑧𝐴𝑧𝐵))

Theoremnfabdw 2979* Bound-variable hypothesis builder for a class abstraction. Version of nfabd 2980 with a disjoint variable condition, which does not require ax-13 2382. (Contributed by Mario Carneiro, 8-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑𝑥{𝑦𝜓})

Theoremnfabd 2980 Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2382. Use the weaker nfabdw 2979 when possible. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-9 2122 and ax-ext 2773. (Revised by Wolf Lammen, 23-May-2023.) (New usage is discouraged.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑𝑥{𝑦𝜓})

Theoremnfabd2 2981 Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2382. (Contributed by Mario Carneiro, 8-Oct-2016.) (Proof shortened by Wolf Lammen, 10-May-2023.) (New usage is discouraged.)
𝑦𝜑    &   ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)       (𝜑𝑥{𝑦𝜓})

Theoremdvelimdc 2982 Deduction form of dvelimc 2983. Usage of this theorem is discouraged because it depends on ax-13 2382. (Contributed by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.)
𝑥𝜑    &   𝑧𝜑    &   (𝜑𝑥𝐴)    &   (𝜑𝑧𝐵)    &   (𝜑 → (𝑧 = 𝑦𝐴 = 𝐵))       (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐵))

Theoremdvelimc 2983 Version of dvelim 2465 for classes. Usage of this theorem is discouraged because it depends on ax-13 2382. (Contributed by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.)
𝑥𝐴    &   𝑧𝐵    &   (𝑧 = 𝑦𝐴 = 𝐵)       (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐵)

Theoremnfcvf 2984 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 2382. See nfcv 2958 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 2773. (Revised by Wolf Lammen, 10-May-2023.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)

Theoremnfcvf2 2985 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 2382. See nfcv 2958 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 = 𝑦𝑦𝑥)

Theoremcleqf 2986 Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions as in dfcleq 2795. See also cleqh 2916. (Contributed by NM, 26-May-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Nov-2019.) Avoid ax-13 2382. (Revised by Wolf Lammen, 10-May-2023.) Avoid ax-10 2143. (Revised by Gino Giotto, 20-Aug-2023.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))

Theoremabid2f 2987 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 2988 Equality of a class variable and a class abstraction. In this version, the fact that 𝑥 is a non-free variable in 𝐴 is explicitly stated as a hypothesis. (Contributed by Thierry Arnoux, 11-May-2017.) Avoid ax-13 2382. (Revised by Wolf Lammen, 13-May-2023.)
𝑥𝐴       (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))

Theoremsbabel 2989* 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

2.1.4.1  Negated equality

Syntaxwne 2990 Extend wff notation to include inequality.
wff 𝐴𝐵

Definitiondf-ne 2991 Define inequality. (Contributed by NM, 26-May-1993.)
(𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)

Theoremneii 2992 Inference associated with df-ne 2991. (Contributed by BJ, 7-Jul-2018.)
𝐴𝐵        ¬ 𝐴 = 𝐵

Theoremneir 2993 Inference associated with df-ne 2991. (Contributed by BJ, 7-Jul-2018.)
¬ 𝐴 = 𝐵       𝐴𝐵

Theoremnne 2994 Negation of inequality. (Contributed by NM, 9-Jun-2006.)
𝐴𝐵𝐴 = 𝐵)

Theoremneneqd 2995 Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝐴𝐵)       (𝜑 → ¬ 𝐴 = 𝐵)

Theoremneneq 2996 From inequality to non-equality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴𝐵 → ¬ 𝐴 = 𝐵)

Theoremneqned 2997 If it is not the case that two classes are equal, then they are unequal. Converse of neneqd 2995. One-way deduction form of df-ne 2991. (Contributed by David Moews, 28-Feb-2017.) Allow a shortening of necon3bi 3016. (Revised by Wolf Lammen, 22-Nov-2019.)
(𝜑 → ¬ 𝐴 = 𝐵)       (𝜑𝐴𝐵)

Theoremneqne 2998 From non-equality to inequality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐴 = 𝐵𝐴𝐵)

Theoremneirr 2999 No class is unequal to itself. Inequality is irreflexive. (Contributed by Stefan O'Rear, 1-Jan-2015.)
¬ 𝐴𝐴

Theoremexmidne 3000 Excluded middle with equality and inequality. (Contributed by NM, 3-Feb-2012.) (Proof shortened by Wolf Lammen, 17-Nov-2019.)
(𝐴 = 𝐵𝐴𝐵)

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