Theorem nfne 2399

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 2307	. 2 |- ( A =/= B <-> -. A = B )
2		nfne.1	. . . 4 |- F/_ x A
3		nfne.2	. . . 4 |- F/_ x B
4	2, 3	nfeq 2287	. . 3 |- F/ x A = B
5	4	nfn 1636	. 2 |- F/ x -. A = B
6	1, 5	nfxfr 1450	1 |- F/ x A =/= B

Syntax hints:   -. wn 3   = wceq 1331   F/wnf 1436   F/_wnfc 2266   =/= wne 2306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307

This theorem is referenced by: (None)