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Theorem nfneld 2430
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 2423 . 2 (𝐴𝐵 ↔ ¬ 𝐴𝐵)
2 nfneld.1 . . . 4 (𝜑𝑥𝐴)
3 nfneld.2 . . . 4 (𝜑𝑥𝐵)
42, 3nfeld 2315 . . 3 (𝜑 → Ⅎ𝑥 𝐴𝐵)
54nfnd 1637 . 2 (𝜑 → Ⅎ𝑥 ¬ 𝐴𝐵)
61, 5nfxfrd 1455 1 (𝜑 → Ⅎ𝑥 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wnf 1440  wcel 2128  wnfc 2286  wnel 2422
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 604  ax-in2 605  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-17 1506  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-fal 1341  df-nf 1441  df-cleq 2150  df-clel 2153  df-nfc 2288  df-nel 2423
This theorem is referenced by: (None)
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