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Theorem nfne 2275
 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 xA
nfne.2 xB
Assertion
Ref Expression
nfne x AB

Proof of Theorem nfne
StepHypRef Expression
1 df-ne 2188 . 2 (AB ↔ ¬ A = B)
2 nfne.1 . . . 4 xA
3 nfne.2 . . . 4 xB
42, 3nfeq 2167 . . 3 x A = B
54nfn 1530 . 2 x ¬ A = B
61, 5nfxfr 1343 1 x AB
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   = wceq 1228  Ⅎwnf 1329  Ⅎwnfc 2147   ≠ wne 2186 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188 This theorem is referenced by: (None)
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