Theorem nfned 2441

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 2348	. 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 2334	. . 3 |- ( ph -> F/ x A = B )
5	4	nfnd 1657	. 2 |- ( ph -> F/ x -. A = B )
6	1, 5	nfxfrd 1475	1 |- ( ph -> F/ x A =/= B )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1353   F/wnf 1460   F/_wnfc 2306   =/= wne 2347

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 614  ax-in2 615  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-cleq 2170  df-nfc 2308  df-ne 2348

This theorem is referenced by: (None)