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

Theorem nfned 2454
Description: Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
Hypotheses
Ref Expression
nfned.1  |-  ( ph  -> 
F/_ x A )
nfned.2  |-  ( ph  -> 
F/_ x B )
Assertion
Ref Expression
nfned  |-  ( ph  ->  F/ x  A  =/= 
B )

Proof of Theorem nfned
StepHypRef Expression
1 df-ne 2361 . 2  |-  ( A  =/=  B  <->  -.  A  =  B )
2 nfned.1 . . . 4  |-  ( ph  -> 
F/_ x A )
3 nfned.2 . . . 4  |-  ( ph  -> 
F/_ x B )
42, 3nfeqd 2347 . . 3  |-  ( ph  ->  F/ x  A  =  B )
54nfnd 1668 . 2  |-  ( ph  ->  F/ x  -.  A  =  B )
61, 5nfxfrd 1486 1  |-  ( ph  ->  F/ x  A  =/= 
B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1364   F/wnf 1471   F/_wnfc 2319    =/= wne 2360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-cleq 2182  df-nfc 2321  df-ne 2361
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator