Theorem nfnel 2410

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
nfnel.1	|- F/_ x A
nfnel.2	|- F/_ x B

Assertion

Ref	Expression
nfnel	|- F/ x A e/ B

Proof of Theorem nfnel

Step	Hyp	Ref	Expression
1		df-nel 2404	. 2 |- ( A e/ B <-> -. A e. B )
2		nfnel.1	. . . 4 |- F/_ x A
3		nfnel.2	. . . 4 |- F/_ x B
4	2, 3	nfel 2290	. . 3 |- F/ x A e. B
5	4	nfn 1636	. 2 |- F/ x -. A e. B
6	1, 5	nfxfr 1450	1 |- F/ x A e/ B

Syntax hints:   -. wn 3   F/wnf 1436   e. wcel 1480   F/_wnfc 2268   e/ wnel 2403

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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-cleq 2132  df-clel 2135  df-nfc 2270  df-nel 2404

This theorem is referenced by: (None)