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Theorem List for Intuitionistic Logic Explorer - 2201-2300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcleqf 2201 Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2137. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 
Theoremabid2f 2202 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 2203* 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
 
Syntaxwne 2204 Extend wff notation to include inequality.
wff 𝐴𝐵
 
Syntaxwnel 2205 Extend wff notation to include negated membership.
wff 𝐴𝐵
 
Definitiondf-ne 2206 Define inequality. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
 
Definitiondf-nel 2207 Define negated membership. (Contributed by NM, 7-Aug-1994.)
(𝐴𝐵 ↔ ¬ 𝐴𝐵)
 
2.1.4.1  Negated equality
 
Theoremneii 2208 Inference associated with df-ne 2206. (Contributed by BJ, 7-Jul-2018.)
𝐴𝐵        ¬ 𝐴 = 𝐵
 
Theoremneir 2209 Inference associated with df-ne 2206. (Contributed by BJ, 7-Jul-2018.)
¬ 𝐴 = 𝐵       𝐴𝐵
 
Theoremnner 2210 Negation of inequality. (Contributed by Jim Kingdon, 23-Dec-2018.)
(𝐴 = 𝐵 → ¬ 𝐴𝐵)
 
Theoremnnedc 2211 Negation of inequality where equality is decidable. (Contributed by Jim Kingdon, 15-May-2018.)
(DECID 𝐴 = 𝐵 → (¬ 𝐴𝐵𝐴 = 𝐵))
 
Theoremdcned 2212 Decidable equality implies decidable negated equality. (Contributed by Jim Kingdon, 3-May-2020.)
(𝜑DECID 𝐴 = 𝐵)       (𝜑DECID 𝐴𝐵)
 
Theoremneirr 2213 No class is unequal to itself. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
¬ 𝐴𝐴
 
Theoremdcne 2214 Decidable equality expressed in terms of . Basically the same as df-dc 743. (Contributed by Jim Kingdon, 14-Mar-2020.)
(DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵𝐴𝐵))
 
Theoremnonconne 2215 Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.)
¬ (𝐴 = 𝐵𝐴𝐵)
 
Theoremneeq1 2216 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
(𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
 
Theoremneeq2 2217 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
(𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
 
Theoremneeq1i 2218 Inference for inequality. (Contributed by NM, 29-Apr-2005.)
𝐴 = 𝐵       (𝐴𝐶𝐵𝐶)
 
Theoremneeq2i 2219 Inference for inequality. (Contributed by NM, 29-Apr-2005.)
𝐴 = 𝐵       (𝐶𝐴𝐶𝐵)
 
Theoremneeq12i 2220 Inference for inequality. (Contributed by NM, 24-Jul-2012.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶𝐵𝐷)
 
Theoremneeq1d 2221 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶𝐵𝐶))
 
Theoremneeq2d 2222 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐶𝐵))
 
Theoremneeq12d 2223 Deduction for inequality. (Contributed by NM, 24-Jul-2012.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶𝐵𝐷))
 
Theoremneneqd 2224 Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝐴𝐵)       (𝜑 → ¬ 𝐴 = 𝐵)
 
Theoremeqnetri 2225 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴 = 𝐵    &   𝐵𝐶       𝐴𝐶
 
Theoremeqnetrd 2226 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremeqnetrri 2227 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴 = 𝐵    &   𝐴𝐶       𝐵𝐶
 
Theoremeqnetrrd 2228 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐴𝐶)       (𝜑𝐵𝐶)
 
Theoremneeqtri 2229 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴𝐵    &   𝐵 = 𝐶       𝐴𝐶
 
Theoremneeqtrd 2230 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(𝜑𝐴𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴𝐶)
 
Theoremneeqtrri 2231 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴𝐵    &   𝐶 = 𝐵       𝐴𝐶
 
Theoremneeqtrrd 2232 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴𝐶)
 
Theoremsyl5eqner 2233 B chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.)
𝐵 = 𝐴    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theorem3netr3d 2234 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
(𝜑𝐴𝐵)    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑𝐶𝐷)
 
Theorem3netr4d 2235 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐴)    &   (𝜑𝐷 = 𝐵)       (𝜑𝐶𝐷)
 
Theorem3netr3g 2236 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
(𝜑𝐴𝐵)    &   𝐴 = 𝐶    &   𝐵 = 𝐷       (𝜑𝐶𝐷)
 
Theorem3netr4g 2237 Substitution of equality into both sides of an inequality. (Contributed by NM, 14-Jun-2012.)
(𝜑𝐴𝐵)    &   𝐶 = 𝐴    &   𝐷 = 𝐵       (𝜑𝐶𝐷)
 
Theoremnecon3abii 2238 Deduction from equality to inequality. (Contributed by NM, 9-Nov-2007.)
(𝐴 = 𝐵𝜑)       (𝐴𝐵 ↔ ¬ 𝜑)
 
Theoremnecon3bbii 2239 Deduction from equality to inequality. (Contributed by NM, 13-Apr-2007.)
(𝜑𝐴 = 𝐵)       𝜑𝐴𝐵)
 
Theoremnecon3bii 2240 Inference from equality to inequality. (Contributed by NM, 23-Feb-2005.)
(𝐴 = 𝐵𝐶 = 𝐷)       (𝐴𝐵𝐶𝐷)
 
Theoremnecon3abid 2241 Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.)
(𝜑 → (𝐴 = 𝐵𝜓))       (𝜑 → (𝐴𝐵 ↔ ¬ 𝜓))
 
Theoremnecon3bbid 2242 Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007.)
(𝜑 → (𝜓𝐴 = 𝐵))       (𝜑 → (¬ 𝜓𝐴𝐵))
 
Theoremnecon3bid 2243 Deduction from equality to inequality. (Contributed by NM, 23-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))       (𝜑 → (𝐴𝐵𝐶𝐷))
 
Theoremnecon3ad 2244 Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
(𝜑 → (𝜓𝐴 = 𝐵))       (𝜑 → (𝐴𝐵 → ¬ 𝜓))
 
Theoremnecon3bd 2245 Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
(𝜑 → (𝐴 = 𝐵𝜓))       (𝜑 → (¬ 𝜓𝐴𝐵))
 
Theoremnecon3d 2246 Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.)
(𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))       (𝜑 → (𝐶𝐷𝐴𝐵))
 
Theoremnesym 2247 Characterization of inequality in terms of reversed equality (see bicom 128). (Contributed by BJ, 7-Jul-2018.)
(𝐴𝐵 ↔ ¬ 𝐵 = 𝐴)
 
Theoremnesymi 2248 Inference associated with nesym 2247. (Contributed by BJ, 7-Jul-2018.)
𝐴𝐵        ¬ 𝐵 = 𝐴
 
Theoremnesymir 2249 Inference associated with nesym 2247. (Contributed by BJ, 7-Jul-2018.)
¬ 𝐴 = 𝐵       𝐵𝐴
 
Theoremnecon3i 2250 Contrapositive inference for inequality. (Contributed by NM, 9-Aug-2006.)
(𝐴 = 𝐵𝐶 = 𝐷)       (𝐶𝐷𝐴𝐵)
 
Theoremnecon3ai 2251 Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
(𝜑𝐴 = 𝐵)       (𝐴𝐵 → ¬ 𝜑)
 
Theoremnecon3bi 2252 Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
(𝐴 = 𝐵𝜑)       𝜑𝐴𝐵)
 
Theoremnecon1aidc 2253 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.)
(DECID 𝜑 → (¬ 𝜑𝐴 = 𝐵))       (DECID 𝜑 → (𝐴𝐵𝜑))
 
Theoremnecon1bidc 2254 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.)
(DECID 𝐴 = 𝐵 → (𝐴𝐵𝜑))       (DECID 𝐴 = 𝐵 → (¬ 𝜑𝐴 = 𝐵))
 
Theoremnecon1idc 2255 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(𝐴𝐵𝐶 = 𝐷)       (DECID 𝐴 = 𝐵 → (𝐶𝐷𝐴 = 𝐵))
 
Theoremnecon2ai 2256 Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.)
(𝐴 = 𝐵 → ¬ 𝜑)       (𝜑𝐴𝐵)
 
Theoremnecon2bi 2257 Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.)
(𝜑𝐴𝐵)       (𝐴 = 𝐵 → ¬ 𝜑)
 
Theoremnecon2i 2258 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)
(𝐴 = 𝐵𝐶𝐷)       (𝐶 = 𝐷𝐴𝐵)
 
Theoremnecon2ad 2259 Contrapositive inference for inequality. (Contributed by NM, 19-Apr-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.)
(𝜑 → (𝐴 = 𝐵 → ¬ 𝜓))       (𝜑 → (𝜓𝐴𝐵))
 
Theoremnecon2bd 2260 Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)
(𝜑 → (𝜓𝐴𝐵))       (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓))
 
Theoremnecon2d 2261 Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.)
(𝜑 → (𝐴 = 𝐵𝐶𝐷))       (𝜑 → (𝐶 = 𝐷𝐴𝐵))
 
Theoremnecon1abiidc 2262 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(DECID 𝜑 → (¬ 𝜑𝐴 = 𝐵))       (DECID 𝜑 → (𝐴𝐵𝜑))
 
Theoremnecon1bbiidc 2263 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(DECID 𝐴 = 𝐵 → (𝐴𝐵𝜑))       (DECID 𝐴 = 𝐵 → (¬ 𝜑𝐴 = 𝐵))
 
Theoremnecon1abiddc 2264 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(𝜑 → (DECID 𝜓 → (¬ 𝜓𝐴 = 𝐵)))       (𝜑 → (DECID 𝜓 → (𝐴𝐵𝜓)))
 
Theoremnecon1bbiddc 2265 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(𝜑 → (DECID 𝐴 = 𝐵 → (𝐴𝐵𝜓)))       (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝜓𝐴 = 𝐵)))
 
Theoremnecon2abiidc 2266 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(DECID 𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝜑))       (DECID 𝜑 → (𝜑𝐴𝐵))
 
Theoremnecon2bbii 2267 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(DECID 𝐴 = 𝐵 → (𝜑𝐴𝐵))       (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝜑))
 
Theoremnecon2abiddc 2268 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(𝜑 → (DECID 𝜓 → (𝐴 = 𝐵 ↔ ¬ 𝜓)))       (𝜑 → (DECID 𝜓 → (𝜓𝐴𝐵)))
 
Theoremnecon2bbiddc 2269 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(𝜑 → (DECID 𝐴 = 𝐵 → (𝜓𝐴𝐵)))       (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝜓)))
 
Theoremnecon4aidc 2270 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(DECID 𝐴 = 𝐵 → (𝐴𝐵 → ¬ 𝜑))       (DECID 𝐴 = 𝐵 → (𝜑𝐴 = 𝐵))
 
Theoremnecon4idc 2271 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(DECID 𝐴 = 𝐵 → (𝐴𝐵𝐶𝐷))       (DECID 𝐴 = 𝐵 → (𝐶 = 𝐷𝐴 = 𝐵))
 
Theoremnecon4addc 2272 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
(𝜑 → (DECID 𝐴 = 𝐵 → (𝐴𝐵 → ¬ 𝜓)))       (𝜑 → (DECID 𝐴 = 𝐵 → (𝜓𝐴 = 𝐵)))
 
Theoremnecon4bddc 2273 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
(𝜑 → (DECID 𝜓 → (¬ 𝜓𝐴𝐵)))       (𝜑 → (DECID 𝜓 → (𝐴 = 𝐵𝜓)))
 
Theoremnecon4ddc 2274 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
(𝜑 → (DECID 𝐴 = 𝐵 → (𝐴𝐵𝐶𝐷)))       (𝜑 → (DECID 𝐴 = 𝐵 → (𝐶 = 𝐷𝐴 = 𝐵)))
 
Theoremnecon4abiddc 2275 Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 18-May-2018.)
(𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝜓 → (𝐴𝐵 ↔ ¬ 𝜓))))       (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝜓 → (𝐴 = 𝐵𝜓))))
 
Theoremnecon4bbiddc 2276 Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
(𝜑 → (DECID 𝜓 → (DECID 𝐴 = 𝐵 → (¬ 𝜓𝐴𝐵))))       (𝜑 → (DECID 𝜓 → (DECID 𝐴 = 𝐵 → (𝜓𝐴 = 𝐵))))
 
Theoremnecon4biddc 2277 Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
(𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → (𝐴𝐵𝐶𝐷))))       (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → (𝐴 = 𝐵𝐶 = 𝐷))))
 
Theoremnecon1addc 2278 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
(𝜑 → (DECID 𝜓 → (¬ 𝜓𝐴 = 𝐵)))       (𝜑 → (DECID 𝜓 → (𝐴𝐵𝜓)))
 
Theoremnecon1bddc 2279 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
(𝜑 → (DECID 𝐴 = 𝐵 → (𝐴𝐵𝜓)))       (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝜓𝐴 = 𝐵)))
 
Theoremnecon1ddc 2280 Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
(𝜑 → (DECID 𝐴 = 𝐵 → (𝐴𝐵𝐶 = 𝐷)))       (𝜑 → (DECID 𝐴 = 𝐵 → (𝐶𝐷𝐴 = 𝐵)))
 
Theoremneneqad 2281 If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2224. One-way deduction form of df-ne 2206. (Contributed by David Moews, 28-Feb-2017.)
(𝜑 → ¬ 𝐴 = 𝐵)       (𝜑𝐴𝐵)
 
Theoremnebidc 2282 Contraposition law for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
(DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → ((𝐴 = 𝐵𝐶 = 𝐷) ↔ (𝐴𝐵𝐶𝐷))))
 
Theorempm13.18 2283 Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
((𝐴 = 𝐵𝐴𝐶) → 𝐵𝐶)
 
Theorempm13.181 2284 Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
((𝐴 = 𝐵𝐵𝐶) → 𝐴𝐶)
 
Theorempm2.21ddne 2285 A contradiction implies anything. Equality/inequality deduction form. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐴𝐵)       (𝜑𝜓)
 
Theoremnecom 2286 Commutation of inequality. (Contributed by NM, 14-May-1999.)
(𝐴𝐵𝐵𝐴)
 
Theoremnecomi 2287 Inference from commutative law for inequality. (Contributed by NM, 17-Oct-2012.)
𝐴𝐵       𝐵𝐴
 
Theoremnecomd 2288 Deduction from commutative law for inequality. (Contributed by NM, 12-Feb-2008.)
(𝜑𝐴𝐵)       (𝜑𝐵𝐴)
 
Theoremneanior 2289 A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)
((𝐴𝐵𝐶𝐷) ↔ ¬ (𝐴 = 𝐵𝐶 = 𝐷))
 
Theoremne3anior 2290 A De Morgan's law for inequality. (Contributed by NM, 30-Sep-2013.) (Proof rewritten by Jim Kingdon, 19-May-2018.)
((𝐴𝐵𝐶𝐷𝐸𝐹) ↔ ¬ (𝐴 = 𝐵𝐶 = 𝐷𝐸 = 𝐹))
 
Theoremnemtbir 2291 An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007.)
𝐴𝐵    &   (𝜑𝐴 = 𝐵)        ¬ 𝜑
 
Theoremnelne1 2292 Two classes are different if they don't contain the same element. (Contributed by NM, 3-Feb-2012.)
((𝐴𝐵 ∧ ¬ 𝐴𝐶) → 𝐵𝐶)
 
Theoremnelne2 2293 Two classes are different if they don't belong to the same class. (Contributed by NM, 25-Jun-2012.)
((𝐴𝐶 ∧ ¬ 𝐵𝐶) → 𝐴𝐵)
 
Theoremnfne 2294 Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥 𝐴𝐵
 
Theoremnfned 2295 Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑 → Ⅎ𝑥 𝐴𝐵)
 
2.1.4.2  Negated membership
 
Theoremneli 2296 Inference associated with df-nel 2207. (Contributed by BJ, 7-Jul-2018.)
𝐴𝐵        ¬ 𝐴𝐵
 
Theoremnelir 2297 Inference associated with df-nel 2207. (Contributed by BJ, 7-Jul-2018.)
¬ 𝐴𝐵       𝐴𝐵
 
Theoremneleq1 2298 Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)
(𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
 
Theoremneleq2 2299 Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)
(𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
 
Theoremneleq12d 2300 Equality theorem for negated membership. (Contributed by FL, 10-Aug-2016.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶𝐵𝐷))
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