Theorem nfned 2349

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 2256	. 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 2243	. . 3 |- ( ph -> F/ x A = B )
5	4	nfnd 1592	. 2 |- ( ph -> F/ x -. A = B )
6	1, 5	nfxfrd 1409	1 |- ( ph -> F/ x A =/= B )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1289   F/wnf 1394   F/_wnfc 2215   =/= wne 2255

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-cleq 2081  df-nfc 2217  df-ne 2256

This theorem is referenced by: (None)