Theorem nfneld 2505

Description: Bound-variable hypothesis builder for negated membership. (Contributed by David Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)

Hypotheses

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

Assertion

Ref	Expression
nfneld	|- ( ph -> F/ x A e/ B )

Proof of Theorem nfneld

Step	Hyp	Ref	Expression
1		df-nel 2498	. 2 |- ( A e/ B <-> -. A e. B )
2		nfneld.1	. . . 4 |- ( ph -> F/_ x A )
3		nfneld.2	. . . 4 |- ( ph -> F/_ x B )
4	2, 3	nfeld 2390	. . 3 |- ( ph -> F/ x A e. B )
5	4	nfnd 1705	. 2 |- ( ph -> F/ x -. A e. B )
6	1, 5	nfxfrd 1523	1 |- ( ph -> F/ x A e/ B )

Syntax hints:   -. wn 3   -> wi 4   F/wnf 1508   e. wcel 2202   F/_wnfc 2361   e/ wnel 2497

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 619  ax-in2 620  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-cleq 2224  df-clel 2227  df-nfc 2363  df-nel 2498

This theorem is referenced by: (None)