Theorem nfnel 2304

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 2207	. 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 2186	. . 3 |- F/ x A e. B
5	4	nfn 1548	. 2 |- F/ x -. A e. B
6	1, 5	nfxfr 1363	1 |- F/ x A e/ B

Syntax hints:   -. wn 3   F/wnf 1349   e. wcel 1393   F/_wnfc 2165   e/ wnel 2205

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036  df-nfc 2167  df-nel 2207

This theorem is referenced by: (None)