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Theorem nfnel 2540
Description: Bound-variable hypothesis builder for inequality. (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 2449 . 2  |-  ( A  e/  B  <->  -.  A  e.  B )
2 nfnel.1 . . . 4  |-  F/_ x A
3 nfnel.2 . . . 4  |-  F/_ x B
42, 3nfel 2427 . . 3  |-  F/ x  A  e.  B
54nfn 1765 . 2  |-  F/ x  -.  A  e.  B
61, 5nfxfr 1557 1  |-  F/ x  A  e/  B
Colors of variables: wff set class
Syntax hints:   -. wn 3   F/wnf 1531    e. wcel 1684   F/_wnfc 2406    e/ wnel 2447
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-cleq 2276  df-clel 2279  df-nfc 2408  df-nel 2449
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