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Theorem nfnel 3034
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
nfnel.1 𝑥𝐴
nfnel.2 𝑥𝐵
Assertion
Ref Expression
nfnel 𝑥 𝐴𝐵

Proof of Theorem nfnel
StepHypRef Expression
1 df-nel 3028 . 2 (𝐴𝐵 ↔ ¬ 𝐴𝐵)
2 nfnel.1 . . . 4 𝑥𝐴
3 nfnel.2 . . . 4 𝑥𝐵
42, 3nfel 2907 . . 3 𝑥 𝐴𝐵
54nfn 1925 . 2 𝑥 ¬ 𝐴𝐵
61, 5nfxfr 1920 1 𝑥 𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wnf 1849  wcel 2131  wnfc 2881  wnel 3027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-cleq 2745  df-clel 2748  df-nfc 2883  df-nel 3028
This theorem is referenced by: (None)
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