Theorem nfned 2298

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 2206	. 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 2192	. . 3 |- ( ph -> F/ x A = B )
5	4	nfnd 1547	. 2 |- ( ph -> F/ x -. A = B )
6	1, 5	nfxfrd 1364	1 |- ( ph -> F/ x A =/= B )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1243   F/wnf 1349   F/_wnfc 2165   =/= wne 2204

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-cleq 2033  df-nfc 2167  df-ne 2206

This theorem is referenced by: (None)