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Theorem List for Intuitionistic Logic Explorer - 2201-2300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem3eltr4d 2201 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐴)    &   (𝜑𝐷 = 𝐵)       (𝜑𝐶𝐷)
 
Theorem3eltr3g 2202 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐴𝐵)    &   𝐴 = 𝐶    &   𝐵 = 𝐷       (𝜑𝐶𝐷)
 
Theorem3eltr4g 2203 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐴𝐵)    &   𝐶 = 𝐴    &   𝐷 = 𝐵       (𝜑𝐶𝐷)
 
Theoremeqeltrid 2204 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
𝐴 = 𝐵    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremeqeltrrid 2205 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
𝐵 = 𝐴    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremeleqtrid 2206 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
𝐴𝐵    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴𝐶)
 
Theoremeleqtrrid 2207 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
𝐴𝐵    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴𝐶)
 
Theoremsyl6eqel 2208 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
(𝜑𝐴 = 𝐵)    &   𝐵𝐶       (𝜑𝐴𝐶)
 
Theoremsyl6eqelr 2209 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
(𝜑𝐵 = 𝐴)    &   𝐵𝐶       (𝜑𝐴𝐶)
 
Theoremeleqtrdi 2210 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
(𝜑𝐴𝐵)    &   𝐵 = 𝐶       (𝜑𝐴𝐶)
 
Theoremeleqtrrdi 2211 A membership and equality inference. (Contributed by NM, 24-Apr-2005.)
(𝜑𝐴𝐵)    &   𝐶 = 𝐵       (𝜑𝐴𝐶)
 
Theoremeleq2s 2212 Substitution of equal classes into a membership antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝐴𝐵𝜑)    &   𝐶 = 𝐵       (𝐴𝐶𝜑)
 
Theoremeqneltrd 2213 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 2214 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 → ¬ 𝐴𝐶)       (𝜑 → ¬ 𝐵𝐶)
 
Theoremneleqtrd 2215 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 2216 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 → ¬ 𝐶𝐵)    &   (𝜑𝐴 = 𝐵)       (𝜑 → ¬ 𝐶𝐴)
 
Theoremcleqh 2217* Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2282. (Contributed by NM, 5-Aug-1993.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)    &   (𝑦𝐵 → ∀𝑥 𝑦𝐵)       (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 
Theoremnelneq 2218 A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997.)
((𝐴𝐶 ∧ ¬ 𝐵𝐶) → ¬ 𝐴 = 𝐵)
 
Theoremnelneq2 2219 A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)
((𝐴𝐵 ∧ ¬ 𝐴𝐶) → ¬ 𝐵 = 𝐶)
 
Theoremeqsb3lem 2220* Lemma for eqsb3 2221. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
([𝑦 / 𝑥]𝑥 = 𝐴𝑦 = 𝐴)
 
Theoremeqsb3 2221* Substitution applied to an atomic wff (class version of equsb3 1902). (Contributed by Rodolfo Medina, 28-Apr-2010.)
([𝑦 / 𝑥]𝑥 = 𝐴𝑦 = 𝐴)
 
Theoremclelsb3 2222* Substitution applied to an atomic wff (class version of elsb3 1929). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)
 
Theoremclelsb4 2223* Substitution applied to an atomic wff (class version of elsb4 1930). (Contributed by Jim Kingdon, 22-Nov-2018.)
([𝑦 / 𝑥]𝐴𝑥𝐴𝑦)
 
Theoremhbxfreq 2224 A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1433 for equivalence version. (Contributed by NM, 21-Aug-2007.)
𝐴 = 𝐵    &   (𝑦𝐵 → ∀𝑥 𝑦𝐵)       (𝑦𝐴 → ∀𝑥 𝑦𝐴)
 
Theoremhblem 2225* Change the free variable of a hypothesis builder. (Contributed by NM, 5-Aug-1993.) (Revised by Andrew Salmon, 11-Jul-2011.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)       (𝑧𝐴 → ∀𝑥 𝑧𝐴)
 
Theoremabeq2 2226* 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 2231 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. 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 set and wff variables not already in the wff. For more information on class variables, see Quine pp. 15-21 and/or Takeuti and Zaring pp. 10-13. (Contributed by NM, 5-Aug-1993.)

(𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
 
Theoremabeq1 2227* Equality of a class variable and a class abstraction. (Contributed by NM, 20-Aug-1993.)
({𝑥𝜑} = 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))
 
Theoremabeq2i 2228 Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 3-Apr-1996.)
𝐴 = {𝑥𝜑}       (𝑥𝐴𝜑)
 
Theoremabeq1i 2229 Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 31-Jul-1994.)
{𝑥𝜑} = 𝐴       (𝜑𝑥𝐴)
 
Theoremabeq2d 2230 Equality of a class variable and a class abstraction (deduction). (Contributed by NM, 16-Nov-1995.)
(𝜑𝐴 = {𝑥𝜓})       (𝜑 → (𝑥𝐴𝜓))
 
Theoremabbi 2231 Equivalent wff's correspond to equal class abstractions. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.)
(∀𝑥(𝜑𝜓) ↔ {𝑥𝜑} = {𝑥𝜓})
 
Theoremabbi2i 2232* Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 5-Aug-1993.)
(𝑥𝐴𝜑)       𝐴 = {𝑥𝜑}
 
Theoremabbii 2233 Equivalent wff's yield equal class abstractions (inference form). (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)       {𝑥𝜑} = {𝑥𝜓}
 
Theoremabbid 2234 Equivalent wff's yield equal class abstractions (deduction form). (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝜓} = {𝑥𝜒})
 
Theoremabbidv 2235* Equivalent wff's yield equal class abstractions (deduction form). (Contributed by NM, 10-Aug-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝜓} = {𝑥𝜒})
 
Theoremabbi2dv 2236* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
(𝜑 → (𝑥𝐴𝜓))       (𝜑𝐴 = {𝑥𝜓})
 
Theoremabbi1dv 2237* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
(𝜑 → (𝜓𝑥𝐴))       (𝜑 → {𝑥𝜓} = 𝐴)
 
Theoremabid2 2238* A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 26-Dec-1993.)
{𝑥𝑥𝐴} = 𝐴
 
Theoremsb8ab 2239 Substitution of variable in class abstraction. (Contributed by Jim Kingdon, 27-Sep-2018.)
𝑦𝜑       {𝑥𝜑} = {𝑦 ∣ [𝑦 / 𝑥]𝜑}
 
Theoremcbvab 2240 Rule used to change bound variables, using implicit substitution. (Contributed by Andrew Salmon, 11-Jul-2011.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝜑} = {𝑦𝜓}
 
Theoremcbvabv 2241* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-May-1999.)
(𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝜑} = {𝑦𝜓}
 
Theoremclelab 2242* Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.)
(𝐴 ∈ {𝑥𝜑} ↔ ∃𝑥(𝑥 = 𝐴𝜑))
 
Theoremclabel 2243* Membership of a class abstraction in another class. (Contributed by NM, 17-Jan-2006.)
({𝑥𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦𝐴 ∧ ∀𝑥(𝑥𝑦𝜑)))
 
Theoremsbab 2244* 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 2245 Extend wff definition to include the not-free predicate for classes.
wff 𝑥𝐴
 
Theoremnfcjust 2246* Justification theorem for df-nfc 2247. (Contributed by Mario Carneiro, 13-Oct-2016.)
(∀𝑦𝑥 𝑦𝐴 ↔ ∀𝑧𝑥 𝑧𝐴)
 
Definitiondf-nfc 2247* 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 quantifier or something that expands to one (such as "there exists at most one"). It is defined in terms of the not-free predicate df-nf 1422 for wffs; see that definition for more information. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
 
Theoremnfci 2248* Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥 𝑦𝐴       𝑥𝐴
 
Theoremnfcii 2249* Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)       𝑥𝐴
 
Theoremnfcr 2250* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝑥𝐴 → Ⅎ𝑥 𝑦𝐴)
 
Theoremnfcrii 2251* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴       (𝑦𝐴 → ∀𝑥 𝑦𝐴)
 
Theoremnfcri 2252* Consequence of the not-free predicate. (Note that unlike nfcr 2250, this does not require 𝑦 and 𝐴 to be disjoint.) (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴       𝑥 𝑦𝐴
 
Theoremnfcd 2253* Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥 𝑦𝐴)       (𝜑𝑥𝐴)
 
Theoremnfceqi 2254 Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝐴 = 𝐵       (𝑥𝐴𝑥𝐵)
 
Theoremnfcxfr 2255 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 2256 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝐴 = 𝐵    &   (𝜑𝑥𝐵)       (𝜑𝑥𝐴)
 
Theoremnfceqdf 2257 An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑥𝜑    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝑥𝐴𝑥𝐵))
 
Theoremnfcv 2258* If 𝑥 is disjoint from 𝐴, then 𝑥 is not free in 𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴
 
Theoremnfcvd 2259* If 𝑥 is disjoint from 𝐴, then 𝑥 is not free in 𝐴. (Contributed by Mario Carneiro, 7-Oct-2016.)
(𝜑𝑥𝐴)
 
Theoremnfab1 2260 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥{𝑥𝜑}
 
Theoremnfnfc1 2261 𝑥 is bound in 𝑥𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝑥𝐴
 
Theoremclelsb3f 2262 Substitution applied to an atomic wff (class version of elsb3 1929). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.)
𝑥𝐴       ([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)
 
Theoremnfab 2263 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑       𝑥{𝑦𝜑}
 
Theoremnfaba1 2264 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑥{𝑦 ∣ ∀𝑥𝜑}
 
Theoremnfnfc 2265 Hypothesis builder for 𝑦𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴       𝑥𝑦𝐴
 
Theoremnfeq 2266 Hypothesis builder for equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥 𝐴 = 𝐵
 
Theoremnfel 2267 Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥 𝐴𝐵
 
Theoremnfeq1 2268* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
𝑥𝐴       𝑥 𝐴 = 𝐵
 
Theoremnfel1 2269* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
𝑥𝐴       𝑥 𝐴𝐵
 
Theoremnfeq2 2270* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
𝑥𝐵       𝑥 𝐴 = 𝐵
 
Theoremnfel2 2271* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
𝑥𝐵       𝑥 𝐴𝐵
 
Theoremnfcrd 2272* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝜑𝑥𝐴)       (𝜑 → Ⅎ𝑥 𝑦𝐴)
 
Theoremnfeqd 2273 Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑 → Ⅎ𝑥 𝐴 = 𝐵)
 
Theoremnfeld 2274 Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑 → Ⅎ𝑥 𝐴𝐵)
 
Theoremdrnfc1 2275 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 2276 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))
 
Theoremnfabd 2277 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑𝑥{𝑦𝜓})
 
Theoremdvelimdc 2278 Deduction form of dvelimc 2279. (Contributed by Mario Carneiro, 8-Oct-2016.)
𝑥𝜑    &   𝑧𝜑    &   (𝜑𝑥𝐴)    &   (𝜑𝑧𝐵)    &   (𝜑 → (𝑧 = 𝑦𝐴 = 𝐵))       (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐵))
 
Theoremdvelimc 2279 Version of dvelim 1970 for classes. (Contributed by Mario Carneiro, 8-Oct-2016.)
𝑥𝐴    &   𝑧𝐵    &   (𝑧 = 𝑦𝐴 = 𝐵)       (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐵)
 
Theoremnfcvf 2280 If 𝑥 and 𝑦 are distinct, then 𝑥 is not free in 𝑦. (Contributed by Mario Carneiro, 8-Oct-2016.)
(¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
 
Theoremnfcvf2 2281 If 𝑥 and 𝑦 are distinct, then 𝑦 is not free in 𝑥. (Contributed by Mario Carneiro, 5-Dec-2016.)
(¬ ∀𝑥 𝑥 = 𝑦𝑦𝑥)
 
Theoremcleqf 2282 Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2217. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 
Theoremabid2f 2283 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.)
𝑥𝐴       {𝑥𝑥𝐴} = 𝐴
 
Theoremsbabel 2284* 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.)
𝑥𝐴       ([𝑦 / 𝑥]{𝑧𝜑} ∈ 𝐴 ↔ {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴)
 
2.1.4  Negated equality and membership
 
2.1.4.1  Negated equality
 
Syntaxwne 2285 Extend wff notation to include inequality.
wff 𝐴𝐵
 
Definitiondf-ne 2286 Define inequality. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
 
Theoremneii 2287 Inference associated with df-ne 2286. (Contributed by BJ, 7-Jul-2018.)
𝐴𝐵        ¬ 𝐴 = 𝐵
 
Theoremneir 2288 Inference associated with df-ne 2286. (Contributed by BJ, 7-Jul-2018.)
¬ 𝐴 = 𝐵       𝐴𝐵
 
Theoremnner 2289 Negation of inequality. (Contributed by Jim Kingdon, 23-Dec-2018.)
(𝐴 = 𝐵 → ¬ 𝐴𝐵)
 
Theoremnnedc 2290 Negation of inequality where equality is decidable. (Contributed by Jim Kingdon, 15-May-2018.)
(DECID 𝐴 = 𝐵 → (¬ 𝐴𝐵𝐴 = 𝐵))
 
Theoremdcned 2291 Decidable equality implies decidable negated equality. (Contributed by Jim Kingdon, 3-May-2020.)
(𝜑DECID 𝐴 = 𝐵)       (𝜑DECID 𝐴𝐵)
 
Theoremneqned 2292 If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2306. One-way deduction form of df-ne 2286. (Contributed by David Moews, 28-Feb-2017.) Allow a shortening of necon3bi 2335. (Revised by Wolf Lammen, 22-Nov-2019.)
(𝜑 → ¬ 𝐴 = 𝐵)       (𝜑𝐴𝐵)
 
Theoremneqne 2293 From non-equality to inequality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐴 = 𝐵𝐴𝐵)
 
Theoremneirr 2294 No class is unequal to itself. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
¬ 𝐴𝐴
 
Theoremeqneqall 2295 A contradiction concerning equality implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
(𝐴 = 𝐵 → (𝐴𝐵𝜑))
 
Theoremdcne 2296 Decidable equality expressed in terms of . Basically the same as df-dc 805. (Contributed by Jim Kingdon, 14-Mar-2020.)
(DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵𝐴𝐵))
 
Theoremnonconne 2297 Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.)
¬ (𝐴 = 𝐵𝐴𝐵)
 
Theoremneeq1 2298 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
(𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
 
Theoremneeq2 2299 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
(𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
 
Theoremneeq1i 2300 Inference for inequality. (Contributed by NM, 29-Apr-2005.)
𝐴 = 𝐵       (𝐴𝐶𝐵𝐶)
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