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Theorem nfned 2276
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 (φxA)
nfned.2 (φxB)
Assertion
Ref Expression
nfned (φ → Ⅎx AB)

Proof of Theorem nfned
StepHypRef Expression
1 df-ne 2188 . 2 (AB ↔ ¬ A = B)
2 nfned.1 . . . 4 (φxA)
3 nfned.2 . . . 4 (φxB)
42, 3nfeqd 2174 . . 3 (φ → Ⅎx A = B)
54nfnd 1529 . 2 (φ → Ⅎx ¬ A = B)
61, 5nfxfrd 1344 1 (φ → Ⅎx AB)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1228  wnf 1329  wnfc 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-5 1316  ax-7 1317  ax-gen 1318  ax-ie2 1364  ax-4 1381  ax-17 1400  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-cleq 2015  df-nfc 2149  df-ne 2188
This theorem is referenced by: (None)
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