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Theorem nfnel 2304
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 2207 . 2 (𝐴𝐵 ↔ ¬ 𝐴𝐵)
2 nfnel.1 . . . 4 𝑥𝐴
3 nfnel.2 . . . 4 𝑥𝐵
42, 3nfel 2186 . . 3 𝑥 𝐴𝐵
54nfn 1548 . 2 𝑥 ¬ 𝐴𝐵
61, 5nfxfr 1363 1 𝑥 𝐴𝐵
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wnf 1349  wcel 1393  wnfc 2165  wnel 2205
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036  df-nfc 2167  df-nel 2207
This theorem is referenced by: (None)
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