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**Theorem List for Metamath Proof Explorer -** 2601-2700 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	drnfc1 2601	Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.)


Theorem	drnfc2 2602	Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.)


Theorem	nfabd2 2603	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)
$/\$

Theorem	nfabd 2604	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)


Theorem	dvelimdc 2605	Deduction form of dvelimc 2606. (Contributed by Mario Carneiro, 8-Oct-2016.)


Theorem	dvelimc 2606	Version of dvelim 2132 for classes. (Contributed by Mario Carneiro, 8-Oct-2016.)


Theorem	nfcvf 2607	If and are distinct, then is not free in . (Contributed by Mario Carneiro, 8-Oct-2016.)


Theorem	nfcvf2 2608	If and are distinct, then is not free in . (Contributed by Mario Carneiro, 5-Dec-2016.)


Theorem	cleqf 2609	Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2535. (Contributed by NM, 26-May-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Nov-2019.)


Theorem	cleqfOLD 2610	Obsolete proof of cleqf 2609 as of 17-Nov-2019. (Contributed by NM, 26-May-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)


Theorem	abid2f 2611	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.)


Theorem	abid2fOLD 2612	Obsolete proof of abid2f 2611 as of 17-Nov-2019. (Contributed by NM, 5-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)


Theorem	abeq2f 2613	Equality of a class variable and a class abstraction. In this version, the fact that is a non-free variable in is explicitely stated as a hypothesis. (Contributed by Thierry Arnoux, 11-May-2017.)


Theorem	sbabel 2614*	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.)


Theorem	sbabelOLD 2615*	Obsolete proof of sbabel 2614 as of 26-Dec-2019. (Contributed by NM, 27-Sep-2003.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)


2.1.4 Negated equality and membership

Syntax	wne 2616	Extend wff notation to include inequality.


Syntax	wnel 2617	Extend wff notation to include negated membership.


Definition	df-ne 2618	Define inequality. (Contributed by NM, 26-May-1993.)


Definition	df-nel 2619	Define negated membership. (Contributed by NM, 7-Aug-1994.)


2.1.4.1 Negated equality

Theorem	neii 2620	Inference associated with df-ne 2618. (Contributed by BJ, 7-Jul-2018.)


Theorem	neir 2621	Inference associated with df-ne 2618. (Contributed by BJ, 7-Jul-2018.)


Theorem	nne 2622	Negation of inequality. (Contributed by NM, 9-Jun-2006.)


Theorem	neneqd 2623	Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)


Theorem	neneq 2624	From inequality to non equality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)


Theorem	neqned 2625	If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2623. One-way deduction form of df-ne 2618. (Contributed by David Moews, 28-Feb-2017.) Allow a shortening of necon3bi 2651. (Revised by Wolf Lammen, 22-Nov-2019.)


Theorem	neirr 2626	No class is unequal to itself. Inequality is irreflexive. (Contributed by Stefan O'Rear, 1-Jan-2015.)


Theorem	exmidne 2627	Excluded middle with equality and inequality. (Contributed by NM, 3-Feb-2012.) (Proof shortened by Wolf Lammen, 17-Nov-2019.)
$\/$

Theorem	exmidneOLD 2628	Obsolete proof of exmidne 2627 as of 17-Nov-2019. (Contributed by NM, 3-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
$\/$

Theorem	eqneqall 2629	A contradiction concerning equality implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)


Theorem	nonconne 2630	Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.) (Proof shortened by Wolf Lammen, 21-Dec-2019.)
$/\$

Theorem	nonconneOLD 2631	Obsolete proof of nonconne 2630 as of 21-Dec-2019. (Contributed by NM, 3-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
$/\$

Theorem	necon3ad 2632	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.)


Theorem	necon3adOLD 2633	Obsolete proof of necon3ad 2632 as of 23-Nov-2019. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)


Theorem	necon3bd 2634	Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)


Theorem	necon2ad 2635	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.)


Theorem	necon2adOLD 2636	Obsolete proof of necon2ad 2635 as of 23-Nov-2019. (Contributed by NM, 19-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)


Theorem	necon2bd 2637	Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)


Theorem	necon1ad 2638	Contrapositive deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Wolf Lammen, 23-Nov-2019.)


Theorem	necon1adOLD 2639	Obsolete proof of necon1ad 2638 as of 23-Nov-2019. (Contributed by NM, 2-Apr-2007.) (Proof modification is discouraged.) (New usage is discouraged.)


Theorem	necon1bd 2640	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.)


Theorem	necon1bdOLD 2641	Obsolete proof of necon1bd 2640 as of 23-Nov-2019. (Contributed by NM, 21-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)


Theorem	necon4ad 2642	Contrapositive inference for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 23-Nov-2019.)


Theorem	necon4adOLD 2643	Obsolete proof of necon4ad 2642 as of 23-Nov-2019. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)


Theorem	necon4bd 2644	Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 23-Nov-2019.)


Theorem	necon4bdOLD 2645	Obsolete proof of necon4bd 2644 as of 23-Nov-2019. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)


Theorem	necon3d 2646	Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.)


Theorem	necon1d 2647	Contrapositive law deduction for inequality. (Contributed by NM, 28-Dec-2008.) (Proof shortened by Andrew Salmon, 25-May-2011.)


Theorem	necon2d 2648	Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.)


Theorem	necon4d 2649	Contrapositive inference for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)


Theorem	necon3ai 2650	Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)


Theorem	necon3bi 2651	Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 22-Nov-2019.)


Theorem	necon3biOLD 2652	Obsolete proof of necon3bi 2651 as of 22-Nov-2019. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)


Theorem	necon1ai 2653	Contrapositive inference for inequality. (Contributed by NM, 12-Feb-2007.) (Proof shortened by Wolf Lammen, 22-Nov-2019.)


Theorem	necon1aiOLD 2654	Obsolete proof of necon1ai 2653 as of 22-Nov-2019. (Contributed by NM, 12-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)


Theorem	necon1bi 2655	Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 22-Nov-2019.)


Theorem	necon1biOLD 2656	Obsolete proof of necon1bi 2655 as of 22-Nov-2019. (Contributed by NM, 18-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)


Theorem	necon2ai 2657	Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 22-Nov-2019.)


Theorem	necon2aiOLD 2658	Obsolete proof of necon2ai 2657 as of 22-Nov-2019. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)


Theorem	necon2bi 2659	Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.)


Theorem	necon4ai 2660	Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 22-Nov-2019.)


Theorem	necon4aiOLD 2661	Obsolete proof of necon4ai 2660 as of 22-Nov-2019. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)


Theorem	necon3i 2662	Contrapositive inference for inequality. (Contributed by NM, 9-Aug-2006.) (Proof shortened by Wolf Lammen, 22-Nov-2019.)


Theorem	necon3iOLD 2663	Obsolete proof of necon3i 2662 as of 22-Nov-2019. (Contributed by NM, 9-Aug-2006.) (Proof modification is discouraged.) (New usage is discouraged.)


Theorem	necon1i 2664	Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)


Theorem	necon2i 2665	Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)


Theorem	necon4i 2666	Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)


Theorem	necon4iOLD 2667	Obsolete proof of necon4i 2666 as of 24-Nov-2019. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)


Theorem	necon3abid 2668	Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.)


Theorem	necon3bbid 2669	Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007.)


Theorem	necon1abid 2670	Contrapositive deduction for inequality. (Contributed by NM, 21-Aug-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)


Theorem	necon1abidOLD 2671	Obsolete proof of necon1abid 2670 as of 24-Nov-2019. (Contributed by NM, 21-Aug-2007.) (Proof modification is discouraged.) (New usage is discouraged.)


Theorem	necon1bbid 2672	Contrapositive inference for inequality. (Contributed by NM, 31-Jan-2008.)


Theorem	necon4abid 2673	Contrapositive law deduction for inequality. (Contributed by NM, 11-Jan-2008.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)


Theorem	necon4abidOLD 2674	Obsolete proof of necon4abid 2673 as of 24-Nov-2019. (Contributed by NM, 11-Jan-2008.) (Proof modification is discouraged.) (New usage is discouraged.)


Theorem	necon4bbid 2675	Contrapositive law deduction for inequality. (Contributed by NM, 9-May-2012.)


Theorem	necon2abid 2676	Contrapositive deduction for inequality. (Contributed by NM, 18-Jul-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)


Theorem	necon2abidOLD 2677	Obsolete proof of necon2abid 2676 as of 24-Nov-2019. (Contributed by NM, 18-Jul-2007.) (Proof modification is discouraged.) (New usage is discouraged.)


Theorem	necon2bbid 2678	Contrapositive deduction for inequality. (Contributed by NM, 13-Apr-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)


Theorem	necon2bbidOLD 2679	Obsolete proof of necon2bbid 2678 as of 24-Nov-2019. (Contributed by NM, 13-Apr-2007.) (Proof modification is discouraged.) (New usage is discouraged.)


Theorem	necon3bid 2680	Deduction from equality to inequality. (Contributed by NM, 23-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)


Theorem	necon4bid 2681	Contrapositive law deduction for inequality. (Contributed by NM, 29-Jun-2007.)


Theorem	necon3abii 2682	Deduction from equality to inequality. (Contributed by NM, 9-Nov-2007.)


Theorem	necon3bbii 2683	Deduction from equality to inequality. (Contributed by NM, 13-Apr-2007.)


Theorem	necon1abii 2684	Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)


Theorem	necon1abiiOLD 2685	Obsolete proof of necon1abii 2684 as of 25-Nov-2019. (Contributed by NM, 17-Mar-2007.) (Proof modification is discouraged.) (New usage is discouraged.)


Theorem	necon1bbii 2686	Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)


Theorem	necon1bbiiOLD 2687	Obsolete proof of necon1bbii 2686 as of 24-Nov-2019. (Contributed by NM, 17-Mar-2007.) (Proof modification is discouraged.) (New usage is discouraged.)


Theorem	necon2abii 2688	Contrapositive inference for inequality. (Contributed by NM, 2-Mar-2007.)


Theorem	necon2bbii 2689	Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)


Theorem	necon3bii 2690	Inference from equality to inequality. (Contributed by NM, 23-Feb-2005.)


Theorem	necom 2691	Commutation of inequality. (Contributed by NM, 14-May-1999.)


Theorem	necomi 2692	Inference from commutative law for inequality. (Contributed by NM, 17-Oct-2012.)


Theorem	necomd 2693	Deduction from commutative law for inequality. (Contributed by NM, 12-Feb-2008.)


Theorem	nesym 2694	Characterization of inequality in terms of reversed equality (see bicom 203). (Contributed by BJ, 7-Jul-2018.)


Theorem	nesymi 2695	Inference associated with nesym 2694. (Contributed by BJ, 7-Jul-2018.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)


Theorem	nesymiOLD 2696	Obsolete proof of nesymi 2695 as of 25-Nov-2019. (Contributed by BJ, 7-Jul-2018.) (Proof modification is discouraged.) (New usage is discouraged.)


Theorem	nesymir 2697	Inference associated with nesym 2694. (Contributed by BJ, 7-Jul-2018.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)


Theorem	nesymirOLD 2698	Obsolete proof of nesymir 2697 as of 25-Nov-2019. (Contributed by BJ, 7-Jul-2018.) (Proof modification is discouraged.) (New usage is discouraged.)


Theorem	neeq1d 2699	Deduction for inequality. (Contributed by NM, 25-Oct-1999.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)


Theorem	neeq2d 2700	Deduction for inequality. (Contributed by NM, 25-Oct-1999.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)

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