Theorem nfned 2494

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

Hypotheses

Ref	Expression
nfned.1	|- ( ph -> F/_ x A )
nfned.2	|- ( ph -> F/_ x B )

Assertion

Ref	Expression
nfned	|- ( ph -> F/ x A =/= B )

Proof of Theorem nfned

Step	Hyp	Ref	Expression
1		df-ne 2401	. 2 |- ( A =/= B <-> -. A = B )
2		nfned.1	. . . 4 |- ( ph -> F/_ x A )
3		nfned.2	. . . 4 |- ( ph -> F/_ x B )
4	2, 3	nfeqd 2387	. . 3 |- ( ph -> F/ x A = B )
5	4	nfnd 1703	. 2 |- ( ph -> F/ x -. A = B )
6	1, 5	nfxfrd 1521	1 |- ( ph -> F/ x A =/= B )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1395   F/wnf 1506   F/_wnfc 2359   =/= wne 2400

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 617  ax-in2 618  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-cleq 2222  df-nfc 2361  df-ne 2401

This theorem is referenced by: (None)