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Theorem nfnel 2694
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
StepHypRef Expression
1 df-nel 2601 . 2  |-  ( A  e/  B  <->  -.  A  e.  B )
2 nfnel.1 . . . 4  |-  F/_ x A
3 nfnel.2 . . . 4  |-  F/_ x B
42, 3nfel 2579 . . 3  |-  F/ x  A  e.  B
54nfn 1811 . 2  |-  F/ x  -.  A  e.  B
61, 5nfxfr 1579 1  |-  F/ x  A  e/  B
Colors of variables: wff set class
Syntax hints:   -. wn 3   F/wnf 1553    e. wcel 1725   F/_wnfc 2558    e/ wnel 2599
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-cleq 2428  df-clel 2431  df-nfc 2560  df-nel 2601
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