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Theorem nfneld 2542
Description: Bound-variable hypothesis builder for inequality. (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 2449 . 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 2434 . . 3  |-  ( ph  ->  F/ x  A  e.  B )
54nfnd 1760 . 2  |-  ( ph  ->  F/ x  -.  A  e.  B )
61, 5nfxfrd 1558 1  |-  ( ph  ->  F/ x  A  e/  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   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-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532  df-cleq 2276  df-clel 2279  df-nfc 2408  df-nel 2449
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