Theorem nfned 2895

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 2795	. 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 2772	. . 3 |- ( ph -> F/ x A = B )
5	4	nfnd 1785	. 2 |- ( ph -> F/ x -. A = B )
6	1, 5	nfxfrd 1780	1 |- ( ph -> F/ x A =/= B )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   F/wnf 1708   F/_wnfc 2751   =/= wne 2794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-cleq 2615  df-nfc 2753  df-ne 2795

This theorem is referenced by: (None)