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Theorem nfneld 2704
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
nfneld.1  |-  ( ph  -> 
F/_ x A )
nfneld.2  |-  ( ph  -> 
F/_ x B )
Assertion
Ref Expression
nfneld  |-  ( ph  ->  F/ x  A  e/  B )

Proof of Theorem nfneld
StepHypRef Expression
1 df-nel 2603 . 2  |-  ( A  e/  B  <->  -.  A  e.  B )
2 nfneld.1 . . . 4  |-  ( ph  -> 
F/_ x A )
3 nfneld.2 . . . 4  |-  ( ph  -> 
F/_ x B )
42, 3nfeld 2588 . . 3  |-  ( ph  ->  F/ x  A  e.  B )
54nfnd 1810 . 2  |-  ( ph  ->  F/ x  -.  A  e.  B )
61, 5nfxfrd 1581 1  |-  ( ph  ->  F/ x  A  e/  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   F/wnf 1554    e. wcel 1726   F/_wnfc 2560    e/ wnel 2601
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555  df-cleq 2430  df-clel 2433  df-nfc 2562  df-nel 2603
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