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Theorem nfne 2440
Description: Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
Hypotheses
Ref Expression
nfne.1  |-  F/_ x A
nfne.2  |-  F/_ x B
Assertion
Ref Expression
nfne  |-  F/ x  A  =/=  B

Proof of Theorem nfne
StepHypRef Expression
1 df-ne 2348 . 2  |-  ( A  =/=  B  <->  -.  A  =  B )
2 nfne.1 . . . 4  |-  F/_ x A
3 nfne.2 . . . 4  |-  F/_ x B
42, 3nfeq 2327 . . 3  |-  F/ x  A  =  B
54nfn 1658 . 2  |-  F/ x  -.  A  =  B
61, 5nfxfr 1474 1  |-  F/ x  A  =/=  B
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1353   F/wnf 1460   F/_wnfc 2306    =/= wne 2347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348
This theorem is referenced by: (None)
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