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Theorem nfned 3120
 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 3017 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 nfned.1 . . . 4 (𝜑𝑥𝐴)
3 nfned.2 . . . 4 (𝜑𝑥𝐵)
42, 3nfeqd 2988 . . 3 (𝜑 → Ⅎ𝑥 𝐴 = 𝐵)
54nfnd 1854 . 2 (𝜑 → Ⅎ𝑥 ¬ 𝐴 = 𝐵)
61, 5nfxfrd 1850 1 (𝜑 → Ⅎ𝑥 𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1533  Ⅎwnf 1780  Ⅎwnfc 2961   ≠ wne 3016 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1777  df-nf 1781  df-cleq 2814  df-nfc 2963  df-ne 3017 This theorem is referenced by: (None)
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