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Theorem nfneld 2731
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 2624 . 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 2599 . . 3  |-  ( ph  ->  F/ x  A  e.  B )
54nfnd 1983 . 2  |-  ( ph  ->  F/ x  -.  A  e.  B )
61, 5nfxfrd 1696 1  |-  ( ph  ->  F/ x  A  e/  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   F/wnf 1666    e. wcel 1886   F/_wnfc 2578    e/ wnel 2622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-ext 2430
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1663  df-nf 1667  df-cleq 2443  df-clel 2446  df-nfc 2580  df-nel 2624
This theorem is referenced by: (None)
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