Theorem nfne 2471

Description: Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)

Hypotheses

Ref	Expression
nfne.1	|- F/_ x A
nfne.2	|- F/_ x B

Assertion

Ref	Expression
nfne	|- F/ x A =/= B

Proof of Theorem nfne

Step	Hyp	Ref	Expression
1		df-ne 2379	. 2 |- ( A =/= B <-> -. A = B )
2		nfne.1	. . . 4 |- F/_ x A
3		nfne.2	. . . 4 |- F/_ x B
4	2, 3	nfeq 2358	. . 3 |- F/ x A = B
5	4	nfn 1682	. 2 |- F/ x -. A = B
6	1, 5	nfxfr 1498	1 |- F/ x A =/= B

Syntax hints:   -. wn 3   = wceq 1373   F/wnf 1484   F/_wnfc 2337   =/= wne 2378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379

This theorem is referenced by: (None)