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Theorem nfned 2882
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 2781 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 nfned.1 . . . 4 (𝜑𝑥𝐴)
3 nfned.2 . . . 4 (𝜑𝑥𝐵)
42, 3nfeqd 2757 . . 3 (𝜑 → Ⅎ𝑥 𝐴 = 𝐵)
54nfnd 1768 . 2 (𝜑 → Ⅎ𝑥 ¬ 𝐴 = 𝐵)
61, 5nfxfrd 1771 1 (𝜑 → Ⅎ𝑥 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1474  wnf 1698  wnfc 2737  wne 2779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-ex 1695  df-nf 1700  df-cleq 2602  df-nfc 2739  df-ne 2781
This theorem is referenced by: (None)
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