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Theorem nfnel 3046
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 3039 . 2 (𝐴𝐵 ↔ ¬ 𝐴𝐵)
2 nfnel.1 . . . 4 𝑥𝐴
3 nfnel.2 . . . 4 𝑥𝐵
42, 3nfel 2909 . . 3 𝑥 𝐴𝐵
54nfn 1852 . 2 𝑥 ¬ 𝐴𝐵
61, 5nfxfr 1847 1 𝑥 𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wnf 1777  wcel 2098  wnfc 2875  wnel 3038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-cleq 2716  df-clel 2802  df-nfc 2877  df-nel 3039
This theorem is referenced by: (None)
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