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

Theorem nnedc 2311
Description: Negation of inequality where equality is decidable. (Contributed by Jim Kingdon, 15-May-2018.)
Assertion
Ref Expression
nnedc  |-  (DECID  A  =  B  ->  ( -.  A  =/=  B  <->  A  =  B ) )

Proof of Theorem nnedc
StepHypRef Expression
1 df-ne 2307 . . . 4  |-  ( A  =/=  B  <->  -.  A  =  B )
21a1i 9 . . 3  |-  (DECID  A  =  B  ->  ( A  =/=  B  <->  -.  A  =  B ) )
32con2biidc 864 . 2  |-  (DECID  A  =  B  ->  ( A  =  B  <->  -.  A  =/=  B ) )
43bicomd 140 1  |-  (DECID  A  =  B  ->  ( -.  A  =/=  B  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104  DECID wdc 819    = wceq 1331    =/= wne 2306
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 603  ax-in2 604  ax-io 698
This theorem depends on definitions:  df-bi 116  df-dc 820  df-ne 2307
This theorem is referenced by:  nn0n0n1ge2b  9123  alzdvds  11541  fzo0dvdseq  11544  algcvgblem  11719
  Copyright terms: Public domain W3C validator