P. 24 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 2301-2400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nfab 2301	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥{𝑦 ∣ 𝜑}
Theorem	nfaba1 2302	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.)
⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥𝜑}
Theorem	nfnfc 2303	Hypothesis builder for Ⅎ𝑦𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥Ⅎ𝑦𝐴
Theorem	nfeq 2304	Hypothesis builder for equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 𝐴 = 𝐵
Theorem	nfel 2305	Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
Theorem	nfeq1 2306*	Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥 𝐴 = 𝐵
Theorem	nfel1 2307*	Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
Theorem	nfeq2 2308*	Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 𝐴 = 𝐵
Theorem	nfel2 2309*	Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
Theorem	nfcrd 2310*	Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)
Theorem	nfeqd 2311	Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥 𝐴 = 𝐵)
Theorem	nfeld 2312	Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ 𝐵)
Theorem	drnfc1 2313	Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.)
⊢ (∀𝑥 𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑦𝐵))
Theorem	drnfc2 2314	Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.)
⊢ (∀𝑥 𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝐴 ↔ Ⅎ𝑧𝐵))
Theorem	nfabdw 2315*	Bound-variable hypothesis builder for a class abstraction. Version of nfabd 2316 with a disjoint variable condition. (Contributed by Mario Carneiro, 8-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})
Theorem	nfabd 2316	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})
Theorem	dvelimdc 2317	Deduction form of dvelimc 2318. (Contributed by Mario Carneiro, 8-Oct-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑧𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑧𝐵)    &   ⊢ (𝜑 → (𝑧 = 𝑦 → 𝐴 = 𝐵))    ⇒   ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵))
Theorem	dvelimc 2318	Version of dvelim 1994 for classes. (Contributed by Mario Carneiro, 8-Oct-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑧𝐵    &   ⊢ (𝑧 = 𝑦 → 𝐴 = 𝐵)    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵)
Theorem	nfcvf 2319	If 𝑥 and 𝑦 are distinct, then 𝑥 is not free in 𝑦. (Contributed by Mario Carneiro, 8-Oct-2016.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
Theorem	nfcvf2 2320	If 𝑥 and 𝑦 are distinct, then 𝑦 is not free in 𝑥. (Contributed by Mario Carneiro, 5-Dec-2016.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥)
Theorem	cleqf 2321	Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2254. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Theorem	abid2f 2322	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.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴
Theorem	sbabel 2323*	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.)
⊢ 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 2324	Extend wff notation to include inequality.
wff 𝐴 ≠ 𝐵
Definition	df-ne 2325	Define inequality. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
Theorem	neii 2326	Inference associated with df-ne 2325. (Contributed by BJ, 7-Jul-2018.)
⊢ 𝐴 ≠ 𝐵    ⇒   ⊢ ¬ 𝐴 = 𝐵
Theorem	neir 2327	Inference associated with df-ne 2325. (Contributed by BJ, 7-Jul-2018.)
⊢ ¬ 𝐴 = 𝐵    ⇒   ⊢ 𝐴 ≠ 𝐵
Theorem	nner 2328	Negation of inequality. (Contributed by Jim Kingdon, 23-Dec-2018.)
⊢ (𝐴 = 𝐵 → ¬ 𝐴 ≠ 𝐵)
Theorem	nnedc 2329	Negation of inequality where equality is decidable. (Contributed by Jim Kingdon, 15-May-2018.)
⊢ (DECID 𝐴 = 𝐵 → (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵))
Theorem	dcned 2330	Decidable equality implies decidable negated equality. (Contributed by Jim Kingdon, 3-May-2020.)
⊢ (𝜑 → DECID 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → DECID 𝐴 ≠ 𝐵)
Theorem	neqned 2331	If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2345. One-way deduction form of df-ne 2325. (Contributed by David Moews, 28-Feb-2017.) Allow a shortening of necon3bi 2374. (Revised by Wolf Lammen, 22-Nov-2019.)
⊢ (𝜑 → ¬ 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐵)
Theorem	neqne 2332	From non-equality to inequality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (¬ 𝐴 = 𝐵 → 𝐴 ≠ 𝐵)
Theorem	neirr 2333	No class is unequal to itself. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
⊢ ¬ 𝐴 ≠ 𝐴
Theorem	eqneqall 2334	A contradiction concerning equality implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝜑))
Theorem	dcne 2335	Decidable equality expressed in terms of ≠. Basically the same as df-dc 821. (Contributed by Jim Kingdon, 14-Mar-2020.)
⊢ (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵))
Theorem	nonconne 2336	Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.)
⊢ ¬ (𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐵)
Theorem	neeq1 2337	Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶))
Theorem	neeq2 2338	Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
⊢ (𝐴 = 𝐵 → (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵))
Theorem	neeq1i 2339	Inference for inequality. (Contributed by NM, 29-Apr-2005.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶)
Theorem	neeq2i 2340	Inference for inequality. (Contributed by NM, 29-Apr-2005.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵)
Theorem	neeq12i 2341	Inference for inequality. (Contributed by NM, 24-Jul-2012.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐷)
Theorem	neeq1d 2342	Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶))
Theorem	neeq2d 2343	Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵))
Theorem	neeq12d 2344	Deduction for inequality. (Contributed by NM, 24-Jul-2012.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐷))
Theorem	neneqd 2345	Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
Theorem	neneq 2346	From inequality to non-equality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 = 𝐵)
Theorem	eqnetri 2347	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐵 ≠ 𝐶    ⇒   ⊢ 𝐴 ≠ 𝐶
Theorem	eqnetrd 2348	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐶)
Theorem	eqnetrri 2349	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐴 ≠ 𝐶    ⇒   ⊢ 𝐵 ≠ 𝐶
Theorem	eqnetrrd 2350	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐶)    ⇒   ⊢ (𝜑 → 𝐵 ≠ 𝐶)
Theorem	neeqtri 2351	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ 𝐴 ≠ 𝐵    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ 𝐴 ≠ 𝐶
Theorem	neeqtrd 2352	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐶)
Theorem	neeqtrri 2353	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ 𝐴 ≠ 𝐵    &   ⊢ 𝐶 = 𝐵    ⇒   ⊢ 𝐴 ≠ 𝐶
Theorem	neeqtrrd 2354	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐶)
Theorem	eqnetrrid 2355	B chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.)
⊢ 𝐵 = 𝐴    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐶)
Theorem	3netr3d 2356	Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐶 ≠ 𝐷)
Theorem	3netr4d 2357	Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐴)    &   ⊢ (𝜑 → 𝐷 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐶 ≠ 𝐷)
Theorem	3netr3g 2358	Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ (𝜑 → 𝐶 ≠ 𝐷)
Theorem	3netr4g 2359	Substitution of equality into both sides of an inequality. (Contributed by NM, 14-Jun-2012.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ 𝐶 = 𝐴    &   ⊢ 𝐷 = 𝐵    ⇒   ⊢ (𝜑 → 𝐶 ≠ 𝐷)
Theorem	necon3abii 2360	Deduction from equality to inequality. (Contributed by NM, 9-Nov-2007.)
⊢ (𝐴 = 𝐵 ↔ 𝜑)    ⇒   ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝜑)
Theorem	necon3bbii 2361	Deduction from equality to inequality. (Contributed by NM, 13-Apr-2007.)
⊢ (𝜑 ↔ 𝐴 = 𝐵)    ⇒   ⊢ (¬ 𝜑 ↔ 𝐴 ≠ 𝐵)
Theorem	necon3bii 2362	Inference from equality to inequality. (Contributed by NM, 23-Feb-2005.)
⊢ (𝐴 = 𝐵 ↔ 𝐶 = 𝐷)    ⇒   ⊢ (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷)
Theorem	necon3abid 2363	Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.)
⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝜓))    ⇒   ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ ¬ 𝜓))
Theorem	necon3bbid 2364	Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007.)
⊢ (𝜑 → (𝜓 ↔ 𝐴 = 𝐵))    ⇒   ⊢ (𝜑 → (¬ 𝜓 ↔ 𝐴 ≠ 𝐵))
Theorem	necon3bid 2365	Deduction from equality to inequality. (Contributed by NM, 23-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷))    ⇒   ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷))
Theorem	necon3ad 2366	Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
⊢ (𝜑 → (𝜓 → 𝐴 = 𝐵))    ⇒   ⊢ (𝜑 → (𝐴 ≠ 𝐵 → ¬ 𝜓))
Theorem	necon3bd 2367	Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
⊢ (𝜑 → (𝐴 = 𝐵 → 𝜓))    ⇒   ⊢ (𝜑 → (¬ 𝜓 → 𝐴 ≠ 𝐵))
Theorem	necon3d 2368	Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.)
⊢ (𝜑 → (𝐴 = 𝐵 → 𝐶 = 𝐷))    ⇒   ⊢ (𝜑 → (𝐶 ≠ 𝐷 → 𝐴 ≠ 𝐵))
Theorem	nesym 2369	Characterization of inequality in terms of reversed equality (see bicom 139). (Contributed by BJ, 7-Jul-2018.)
⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐵 = 𝐴)
Theorem	nesymi 2370	Inference associated with nesym 2369. (Contributed by BJ, 7-Jul-2018.)
⊢ 𝐴 ≠ 𝐵    ⇒   ⊢ ¬ 𝐵 = 𝐴
Theorem	nesymir 2371	Inference associated with nesym 2369. (Contributed by BJ, 7-Jul-2018.)
⊢ ¬ 𝐴 = 𝐵    ⇒   ⊢ 𝐵 ≠ 𝐴
Theorem	necon3i 2372	Contrapositive inference for inequality. (Contributed by NM, 9-Aug-2006.)
⊢ (𝐴 = 𝐵 → 𝐶 = 𝐷)    ⇒   ⊢ (𝐶 ≠ 𝐷 → 𝐴 ≠ 𝐵)
Theorem	necon3ai 2373	Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝐴 ≠ 𝐵 → ¬ 𝜑)
Theorem	necon3bi 2374	Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
⊢ (𝐴 = 𝐵 → 𝜑)    ⇒   ⊢ (¬ 𝜑 → 𝐴 ≠ 𝐵)
Theorem	necon1aidc 2375	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.)
⊢ (DECID 𝜑 → (¬ 𝜑 → 𝐴 = 𝐵))    ⇒   ⊢ (DECID 𝜑 → (𝐴 ≠ 𝐵 → 𝜑))
Theorem	necon1bidc 2376	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.)
⊢ (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝜑))    ⇒   ⊢ (DECID 𝐴 = 𝐵 → (¬ 𝜑 → 𝐴 = 𝐵))
Theorem	necon1idc 2377	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
⊢ (𝐴 ≠ 𝐵 → 𝐶 = 𝐷)    ⇒   ⊢ (DECID 𝐴 = 𝐵 → (𝐶 ≠ 𝐷 → 𝐴 = 𝐵))
Theorem	necon2ai 2378	Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.)
⊢ (𝐴 = 𝐵 → ¬ 𝜑)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐵)
Theorem	necon2bi 2379	Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    ⇒   ⊢ (𝐴 = 𝐵 → ¬ 𝜑)
Theorem	necon2i 2380	Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)
⊢ (𝐴 = 𝐵 → 𝐶 ≠ 𝐷)    ⇒   ⊢ (𝐶 = 𝐷 → 𝐴 ≠ 𝐵)
Theorem	necon2ad 2381	Contrapositive inference for inequality. (Contributed by NM, 19-Apr-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.)
⊢ (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓))    ⇒   ⊢ (𝜑 → (𝜓 → 𝐴 ≠ 𝐵))
Theorem	necon2bd 2382	Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)
⊢ (𝜑 → (𝜓 → 𝐴 ≠ 𝐵))    ⇒   ⊢ (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓))
Theorem	necon2d 2383	Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.)
⊢ (𝜑 → (𝐴 = 𝐵 → 𝐶 ≠ 𝐷))    ⇒   ⊢ (𝜑 → (𝐶 = 𝐷 → 𝐴 ≠ 𝐵))
Theorem	necon1abiidc 2384	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
⊢ (DECID 𝜑 → (¬ 𝜑 ↔ 𝐴 = 𝐵))    ⇒   ⊢ (DECID 𝜑 → (𝐴 ≠ 𝐵 ↔ 𝜑))
Theorem	necon1bbiidc 2385	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
⊢ (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 ↔ 𝜑))    ⇒   ⊢ (DECID 𝐴 = 𝐵 → (¬ 𝜑 ↔ 𝐴 = 𝐵))
Theorem	necon1abiddc 2386	Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 ↔ 𝐴 = 𝐵)))    ⇒   ⊢ (𝜑 → (DECID 𝜓 → (𝐴 ≠ 𝐵 ↔ 𝜓)))
Theorem	necon1bbiddc 2387	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 ↔ 𝜓)))    ⇒   ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝜓 ↔ 𝐴 = 𝐵)))
Theorem	necon2abiidc 2388	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
⊢ (DECID 𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝜑))    ⇒   ⊢ (DECID 𝜑 → (𝜑 ↔ 𝐴 ≠ 𝐵))
Theorem	necon2bbiidc 2389	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
⊢ (DECID 𝐴 = 𝐵 → (𝜑 ↔ 𝐴 ≠ 𝐵))    ⇒   ⊢ (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝜑))
Theorem	necon2abiddc 2390	Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
⊢ (𝜑 → (DECID 𝜓 → (𝐴 = 𝐵 ↔ ¬ 𝜓)))    ⇒   ⊢ (𝜑 → (DECID 𝜓 → (𝜓 ↔ 𝐴 ≠ 𝐵)))
Theorem	necon2bbiddc 2391	Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝜓 ↔ 𝐴 ≠ 𝐵)))    ⇒   ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝜓)))
Theorem	necon4aidc 2392	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
⊢ (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → ¬ 𝜑))    ⇒   ⊢ (DECID 𝐴 = 𝐵 → (𝜑 → 𝐴 = 𝐵))
Theorem	necon4idc 2393	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
⊢ (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝐶 ≠ 𝐷))    ⇒   ⊢ (DECID 𝐴 = 𝐵 → (𝐶 = 𝐷 → 𝐴 = 𝐵))
Theorem	necon4addc 2394	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → ¬ 𝜓)))    ⇒   ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝜓 → 𝐴 = 𝐵)))
Theorem	necon4bddc 2395	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 → 𝐴 ≠ 𝐵)))    ⇒   ⊢ (𝜑 → (DECID 𝜓 → (𝐴 = 𝐵 → 𝜓)))
Theorem	necon4ddc 2396	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝐶 ≠ 𝐷)))    ⇒   ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐶 = 𝐷 → 𝐴 = 𝐵)))
Theorem	necon4abiddc 2397	Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 18-May-2018.)
⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝜓 → (𝐴 ≠ 𝐵 ↔ ¬ 𝜓))))    ⇒   ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝜓 → (𝐴 = 𝐵 ↔ 𝜓))))
Theorem	necon4bbiddc 2398	Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
⊢ (𝜑 → (DECID 𝜓 → (DECID 𝐴 = 𝐵 → (¬ 𝜓 ↔ 𝐴 ≠ 𝐵))))    ⇒   ⊢ (𝜑 → (DECID 𝜓 → (DECID 𝐴 = 𝐵 → (𝜓 ↔ 𝐴 = 𝐵))))
Theorem	necon4biddc 2399	Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷))))    ⇒   ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷))))
Theorem	necon1addc 2400	Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 → 𝐴 = 𝐵)))    ⇒   ⊢ (𝜑 → (DECID 𝜓 → (𝐴 ≠ 𝐵 → 𝜓)))