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Theorem xorbin 1379
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 1371 . . 3  |-  ( (
ph  \/_  ps )  <->  ( ( ph  \/  ps )  /\  -.  ( ph  /\ 
ps ) ) )
2 imnan 685 . . . . 5  |-  ( (
ph  ->  -.  ps )  <->  -.  ( ph  /\  ps ) )
32biimpri 132 . . . 4  |-  ( -.  ( ph  /\  ps )  ->  ( ph  ->  -. 
ps ) )
43adantl 275 . . 3  |-  ( ( ( ph  \/  ps )  /\  -.  ( ph  /\ 
ps ) )  -> 
( ph  ->  -.  ps ) )
51, 4sylbi 120 . 2  |-  ( (
ph  \/_  ps )  ->  ( ph  ->  -.  ps ) )
6 pm2.53 717 . . . . 5  |-  ( ( ps  \/  ph )  ->  ( -.  ps  ->  ph ) )
76orcoms 725 . . . 4  |-  ( (
ph  \/  ps )  ->  ( -.  ps  ->  ph ) )
87adantr 274 . . 3  |-  ( ( ( ph  \/  ps )  /\  -.  ( ph  /\ 
ps ) )  -> 
( -.  ps  ->  ph ) )
91, 8sylbi 120 . 2  |-  ( (
ph  \/_  ps )  ->  ( -.  ps  ->  ph ) )
105, 9impbid 128 1  |-  ( (
ph  \/_  ps )  ->  ( ph  <->  -.  ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 703    \/_ wxo 1370
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 609  ax-in2 610  ax-io 704
This theorem depends on definitions:  df-bi 116  df-xor 1371
This theorem is referenced by:  xornbi  1381  zeo4  11822  odd2np1  11825
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