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Theorem nfned 3044
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 2944 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 nfned.1 . . . 4 (𝜑𝑥𝐴)
3 nfned.2 . . . 4 (𝜑𝑥𝐵)
42, 3nfeqd 2921 . . 3 (𝜑 → Ⅎ𝑥 𝐴 = 𝐵)
54nfnd 1936 . 2 (𝜑 → Ⅎ𝑥 ¬ 𝐴 = 𝐵)
61, 5nfxfrd 1930 1 (𝜑 → Ⅎ𝑥 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1631  wnf 1856  wnfc 2900  wne 2943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-ex 1853  df-nf 1858  df-cleq 2764  df-nfc 2902  df-ne 2944
This theorem is referenced by: (None)
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