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Theorem nfned 2379
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 2286 . 2  |-  ( A  =/=  B  <->  -.  A  =  B )
2 nfned.1 . . . 4  |-  ( ph  -> 
F/_ x A )
3 nfned.2 . . . 4  |-  ( ph  -> 
F/_ x B )
42, 3nfeqd 2273 . . 3  |-  ( ph  ->  F/ x  A  =  B )
54nfnd 1620 . 2  |-  ( ph  ->  F/ x  -.  A  =  B )
61, 5nfxfrd 1436 1  |-  ( ph  ->  F/ x  A  =/= 
B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1316   F/wnf 1421   F/_wnfc 2245    =/= wne 2285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie2 1455  ax-4 1472  ax-17 1491  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-cleq 2110  df-nfc 2247  df-ne 2286
This theorem is referenced by: (None)
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