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

Theorem xorbin 1316
Description: A consequence of exclusive or. In classical logic the converse also holds. (Contributed by Jim Kingdon, 8-Mar-2018.)
Assertion
Ref Expression
xorbin  |-  ( (
ph  \/_  ps )  ->  ( ph  <->  -.  ps )
)

Proof of Theorem xorbin
StepHypRef Expression
1 df-xor 1308 . . 3  |-  ( (
ph  \/_  ps )  <->  ( ( ph  \/  ps )  /\  -.  ( ph  /\ 
ps ) ) )
2 imnan 657 . . . . 5  |-  ( (
ph  ->  -.  ps )  <->  -.  ( ph  /\  ps ) )
32biimpri 131 . . . 4  |-  ( -.  ( ph  /\  ps )  ->  ( ph  ->  -. 
ps ) )
43adantl 271 . . 3  |-  ( ( ( ph  \/  ps )  /\  -.  ( ph  /\ 
ps ) )  -> 
( ph  ->  -.  ps ) )
51, 4sylbi 119 . 2  |-  ( (
ph  \/_  ps )  ->  ( ph  ->  -.  ps ) )
6 pm2.53 674 . . . . 5  |-  ( ( ps  \/  ph )  ->  ( -.  ps  ->  ph ) )
76orcoms 682 . . . 4  |-  ( (
ph  \/  ps )  ->  ( -.  ps  ->  ph ) )
87adantr 270 . . 3  |-  ( ( ( ph  \/  ps )  /\  -.  ( ph  /\ 
ps ) )  -> 
( -.  ps  ->  ph ) )
91, 8sylbi 119 . 2  |-  ( (
ph  \/_  ps )  ->  ( -.  ps  ->  ph ) )
105, 9impbid 127 1  |-  ( (
ph  \/_  ps )  ->  ( ph  <->  -.  ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662    \/_ wxo 1307
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-xor 1308
This theorem is referenced by:  xornbi  1318  zeo4  10414  odd2np1  10417
  Copyright terms: Public domain W3C validator