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Theorem nfnel 2411
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 2405 . 2  |-  ( A  e/  B  <->  -.  A  e.  B )
2 nfnel.1 . . . 4  |-  F/_ x A
3 nfnel.2 . . . 4  |-  F/_ x B
42, 3nfel 2291 . . 3  |-  F/ x  A  e.  B
54nfn 1637 . 2  |-  F/ x  -.  A  e.  B
61, 5nfxfr 1451 1  |-  F/ x  A  e/  B
Colors of variables: wff set class
Syntax hints:   -. wn 3   F/wnf 1437    e. wcel 1481   F/_wnfc 2269    e/ wnel 2404
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-cleq 2133  df-clel 2136  df-nfc 2271  df-nel 2405
This theorem is referenced by: (None)
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