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Theorem excxor 1339
Description: This tautology shows that xor is really exclusive. (Contributed by FL, 22-Nov-2010.) (Proof rewritten by Jim Kingdon, 5-May-2018.)
Assertion
Ref Expression
excxor  |-  ( (
ph  \/_  ps )  <->  ( ( ph  /\  -.  ps )  \/  ( -.  ph  /\  ps )
) )

Proof of Theorem excxor
StepHypRef Expression
1 xoranor 1338 . . 3  |-  ( (
ph  \/_  ps )  <->  ( ( ph  \/  ps )  /\  ( -.  ph  \/  -.  ps ) ) )
2 andi 790 . . 3  |-  ( ( ( ph  \/  ps )  /\  ( -.  ph  \/  -.  ps ) )  <-> 
( ( ( ph  \/  ps )  /\  -.  ph )  \/  ( (
ph  \/  ps )  /\  -.  ps ) ) )
3 orcom 700 . . . . 5  |-  ( ( ( ps  /\  -.  ph )  \/  ( ph  /\ 
-.  ph ) )  <->  ( ( ph  /\  -.  ph )  \/  ( ps  /\  -.  ph ) ) )
4 pm3.24 665 . . . . . 6  |-  -.  ( ph  /\  -.  ph )
54biorfi 718 . . . . 5  |-  ( ( ps  /\  -.  ph ) 
<->  ( ( ps  /\  -.  ph )  \/  ( ph  /\  -.  ph )
) )
6 andir 791 . . . . 5  |-  ( ( ( ph  \/  ps )  /\  -.  ph )  <->  ( ( ph  /\  -.  ph )  \/  ( ps 
/\  -.  ph ) ) )
73, 5, 63bitr4ri 212 . . . 4  |-  ( ( ( ph  \/  ps )  /\  -.  ph )  <->  ( ps  /\  -.  ph ) )
8 pm5.61 766 . . . 4  |-  ( ( ( ph  \/  ps )  /\  -.  ps )  <->  (
ph  /\  -.  ps )
)
97, 8orbi12i 736 . . 3  |-  ( ( ( ( ph  \/  ps )  /\  -.  ph )  \/  ( ( ph  \/  ps )  /\  -.  ps ) )  <->  ( ( ps  /\  -.  ph )  \/  ( ph  /\  -.  ps ) ) )
101, 2, 93bitri 205 . 2  |-  ( (
ph  \/_  ps )  <->  ( ( ps  /\  -.  ph )  \/  ( ph  /\ 
-.  ps ) ) )
11 orcom 700 . 2  |-  ( ( ( ps  /\  -.  ph )  \/  ( ph  /\ 
-.  ps ) )  <->  ( ( ph  /\  -.  ps )  \/  ( ps  /\  -.  ph ) ) )
12 ancom 264 . . 3  |-  ( ( ps  /\  -.  ph ) 
<->  ( -.  ph  /\  ps ) )
1312orbi2i 734 . 2  |-  ( ( ( ph  /\  -.  ps )  \/  ( ps  /\  -.  ph )
)  <->  ( ( ph  /\ 
-.  ps )  \/  ( -.  ph  /\  ps )
) )
1410, 11, 133bitri 205 1  |-  ( (
ph  \/_  ps )  <->  ( ( ph  /\  -.  ps )  \/  ( -.  ph  /\  ps )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    <-> wb 104    \/ wo 680    \/_ wxo 1336
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 586  ax-in2 587  ax-io 681
This theorem depends on definitions:  df-bi 116  df-xor 1337
This theorem is referenced by:  xordc  1353  symdifxor  3310
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