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

Theorem nfnel 2647
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  |-  F/_ x A
nfnel.2  |-  F/_ x B
Assertion
Ref Expression
nfnel  |-  F/ x  A  e/  B

Proof of Theorem nfnel
StepHypRef Expression
1 df-nel 2554 . 2  |-  ( A  e/  B  <->  -.  A  e.  B )
2 nfnel.1 . . . 4  |-  F/_ x A
3 nfnel.2 . . . 4  |-  F/_ x B
42, 3nfel 2532 . . 3  |-  F/ x  A  e.  B
54nfn 1801 . 2  |-  F/ x  -.  A  e.  B
61, 5nfxfr 1576 1  |-  F/ x  A  e/  B
Colors of variables: wff set class
Syntax hints:   -. wn 3   F/wnf 1550    e. wcel 1717   F/_wnfc 2511    e/ wnel 2552
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-cleq 2381  df-clel 2384  df-nfc 2513  df-nel 2554
  Copyright terms: Public domain W3C validator