Theorem nfneld 2409

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 2402	. 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 2295	. . 3 |- ( ph -> F/ x A e. B )
5	4	nfnd 1635	. 2 |- ( ph -> F/ x -. A e. B )
6	1, 5	nfxfrd 1451	1 |- ( ph -> F/ x A e/ B )

Syntax hints:   -. wn 3   -> wi 4   F/wnf 1436   e. wcel 1480   F/_wnfc 2266   e/ wnel 2401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-cleq 2130  df-clel 2133  df-nfc 2268  df-nel 2402

This theorem is referenced by: (None)