ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfneld Unicode version

Theorem nfneld 2305
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
nfneld.1  |-  ( ph  -> 
F/_ x A )
nfneld.2  |-  ( ph  -> 
F/_ x B )
Assertion
Ref Expression
nfneld  |-  ( ph  ->  F/ x  A  e/  B )

Proof of Theorem nfneld
StepHypRef Expression
1 df-nel 2207 . 2  |-  ( A  e/  B  <->  -.  A  e.  B )
2 nfneld.1 . . . 4  |-  ( ph  -> 
F/_ x A )
3 nfneld.2 . . . 4  |-  ( ph  -> 
F/_ x B )
42, 3nfeld 2193 . . 3  |-  ( ph  ->  F/ x  A  e.  B )
54nfnd 1547 . 2  |-  ( ph  ->  F/ x  -.  A  e.  B )
61, 5nfxfrd 1364 1  |-  ( ph  ->  F/ x  A  e/  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   F/wnf 1349    e. wcel 1393   F/_wnfc 2165    e/ 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-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  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-cleq 2033  df-clel 2036  df-nfc 2167  df-nel 2207
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator