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Theorem nfneld 3054
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 3047 . 2 (𝐴𝐵 ↔ ¬ 𝐴𝐵)
2 nfneld.1 . . . 4 (𝜑𝑥𝐴)
3 nfneld.2 . . . 4 (𝜑𝑥𝐵)
42, 3nfeld 2915 . . 3 (𝜑 → Ⅎ𝑥 𝐴𝐵)
54nfnd 1862 . 2 (𝜑 → Ⅎ𝑥 ¬ 𝐴𝐵)
61, 5nfxfrd 1857 1 (𝜑 → Ⅎ𝑥 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wnf 1786  wcel 2107  wnfc 2884  wnel 3046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783  df-nf 1787  df-cleq 2725  df-clel 2811  df-nfc 2886  df-nel 3047
This theorem is referenced by: (None)
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