MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfnel Structured version   Visualization version   GIF version

Theorem nfnel 2889
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 2782 . 2 (𝐴𝐵 ↔ ¬ 𝐴𝐵)
2 nfnel.1 . . . 4 𝑥𝐴
3 nfnel.2 . . . 4 𝑥𝐵
42, 3nfel 2762 . . 3 𝑥 𝐴𝐵
54nfn 1767 . 2 𝑥 ¬ 𝐴𝐵
61, 5nfxfr 1770 1 𝑥 𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wnf 1698  wcel 1976  wnfc 2737  wnel 2780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-cleq 2602  df-clel 2605  df-nfc 2739  df-nel 2782
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator