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Theorem nfnel 2411
 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 2405 . 2 (𝐴𝐵 ↔ ¬ 𝐴𝐵)
2 nfnel.1 . . . 4 𝑥𝐴
3 nfnel.2 . . . 4 𝑥𝐵
42, 3nfel 2291 . . 3 𝑥 𝐴𝐵
54nfn 1637 . 2 𝑥 ¬ 𝐴𝐵
61, 5nfxfr 1451 1 𝑥 𝐴𝐵
 Colors of variables: wff set class Syntax hints:  ¬ wn 3  Ⅎwnf 1437   ∈ wcel 1481  Ⅎwnfc 2269   ∉ wnel 2404 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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-cleq 2133  df-clel 2136  df-nfc 2271  df-nel 2405 This theorem is referenced by: (None)
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