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Theorem nfneld 2443
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 2436 . 2 (𝐴𝐵 ↔ ¬ 𝐴𝐵)
2 nfneld.1 . . . 4 (𝜑𝑥𝐴)
3 nfneld.2 . . . 4 (𝜑𝑥𝐵)
42, 3nfeld 2328 . . 3 (𝜑 → Ⅎ𝑥 𝐴𝐵)
54nfnd 1650 . 2 (𝜑 → Ⅎ𝑥 ¬ 𝐴𝐵)
61, 5nfxfrd 1468 1 (𝜑 → Ⅎ𝑥 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wnf 1453  wcel 2141  wnfc 2299  wnel 2435
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 609  ax-in2 610  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-cleq 2163  df-clel 2166  df-nfc 2301  df-nel 2436
This theorem is referenced by: (None)
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