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Theorem nfned 2644
Description: Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
Hypotheses
Ref Expression
nfned.1  |-  ( ph  -> 
F/_ x A )
nfned.2  |-  ( ph  -> 
F/_ x B )
Assertion
Ref Expression
nfned  |-  ( ph  ->  F/ x  A  =/= 
B )

Proof of Theorem nfned
StepHypRef Expression
1 df-ne 2554 . 2  |-  ( A  =/=  B  <->  -.  A  =  B )
2 nfned.1 . . . 4  |-  ( ph  -> 
F/_ x A )
3 nfned.2 . . . 4  |-  ( ph  -> 
F/_ x B )
42, 3nfeqd 2539 . . 3  |-  ( ph  ->  F/ x  A  =  B )
54nfnd 1799 . 2  |-  ( ph  ->  F/ x  -.  A  =  B )
61, 5nfxfrd 1577 1  |-  ( ph  ->  F/ x  A  =/= 
B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   F/wnf 1550    = wceq 1649   F/_wnfc 2512    =/= wne 2552
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551  df-cleq 2382  df-nfc 2514  df-ne 2554
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