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Theorem nfned 2554
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 2461 . 2  |-  ( A  =/=  B  <->  -.  A  =  B )
2 nfned.1 . . . 4  |-  ( ph  -> 
F/_ x A )
3 nfned.2 . . . 4  |-  ( ph  -> 
F/_ x B )
42, 3nfeqd 2446 . . 3  |-  ( ph  ->  F/ x  A  =  B )
54nfnd 1772 . 2  |-  ( ph  ->  F/ x  -.  A  =  B )
61, 5nfxfrd 1561 1  |-  ( ph  ->  F/ x  A  =/= 
B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   F/wnf 1534    = wceq 1632   F/_wnfc 2419    =/= wne 2459
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-nf 1535  df-cleq 2289  df-nfc 2421  df-ne 2461
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