ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfned GIF version

Theorem nfned 2379
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 2286 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 nfned.1 . . . 4 (𝜑𝑥𝐴)
3 nfned.2 . . . 4 (𝜑𝑥𝐵)
42, 3nfeqd 2273 . . 3 (𝜑 → Ⅎ𝑥 𝐴 = 𝐵)
54nfnd 1620 . 2 (𝜑 → Ⅎ𝑥 ¬ 𝐴 = 𝐵)
61, 5nfxfrd 1436 1 (𝜑 → Ⅎ𝑥 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1316  wnf 1421  wnfc 2245  wne 2285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie2 1455  ax-4 1472  ax-17 1491  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-cleq 2110  df-nfc 2247  df-ne 2286
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator