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

Theorem nfneld 2358
Description: Bound-variable hypothesis builder for negated membership. (Contributed by David Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
Hypotheses
Ref Expression
nfneld.1 (𝜑𝑥𝐴)
nfneld.2 (𝜑𝑥𝐵)
Assertion
Ref Expression
nfneld (𝜑 → Ⅎ𝑥 𝐴𝐵)

Proof of Theorem nfneld
StepHypRef Expression
1 df-nel 2351 . 2 (𝐴𝐵 ↔ ¬ 𝐴𝐵)
2 nfneld.1 . . . 4 (𝜑𝑥𝐴)
3 nfneld.2 . . . 4 (𝜑𝑥𝐵)
42, 3nfeld 2244 . . 3 (𝜑 → Ⅎ𝑥 𝐴𝐵)
54nfnd 1592 . 2 (𝜑 → Ⅎ𝑥 ¬ 𝐴𝐵)
61, 5nfxfrd 1409 1 (𝜑 → Ⅎ𝑥 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wnf 1394  wcel 1438  wnfc 2215  wnel 2350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-cleq 2081  df-clel 2084  df-nfc 2217  df-nel 2351
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator