Theorem nfnel 2459

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 2453	. 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 2338	. . 3 |- F/ x A e. B
5	4	nfn 1668	. 2 |- F/ x -. A e. B
6	1, 5	nfxfr 1484	1 |- F/ x A e/ B

Syntax hints:   -. wn 3   F/wnf 1470   e. wcel 2158   F/_wnfc 2316   e/ wnel 2452

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-cleq 2180  df-clel 2183  df-nfc 2318  df-nel 2453

This theorem is referenced by: (None)