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Theorem excxor 1285
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 1284 . . 3  |-  ( (
ph  \/_  ps )  <->  ( ( ph  \/  ps )  /\  ( -.  ph  \/  -.  ps ) ) )
2 andi 742 . . 3  |-  ( ( ( ph  \/  ps )  /\  ( -.  ph  \/  -.  ps ) )  <-> 
( ( ( ph  \/  ps )  /\  -.  ph )  \/  ( (
ph  \/  ps )  /\  -.  ps ) ) )
3 orcom 657 . . . . 5  |-  ( ( ( ps  /\  -.  ph )  \/  ( ph  /\ 
-.  ph ) )  <->  ( ( ph  /\  -.  ph )  \/  ( ps  /\  -.  ph ) ) )
4 pm3.24 637 . . . . . 6  |-  -.  ( ph  /\  -.  ph )
54biorfi 675 . . . . 5  |-  ( ( ps  /\  -.  ph ) 
<->  ( ( ps  /\  -.  ph )  \/  ( ph  /\  -.  ph )
) )
6 andir 743 . . . . 5  |-  ( ( ( ph  \/  ps )  /\  -.  ph )  <->  ( ( ph  /\  -.  ph )  \/  ( ps 
/\  -.  ph ) ) )
73, 5, 63bitr4ri 206 . . . 4  |-  ( ( ( ph  \/  ps )  /\  -.  ph )  <->  ( ps  /\  -.  ph ) )
8 pm5.61 718 . . . 4  |-  ( ( ( ph  \/  ps )  /\  -.  ps )  <->  (
ph  /\  -.  ps )
)
97, 8orbi12i 691 . . 3  |-  ( ( ( ( ph  \/  ps )  /\  -.  ph )  \/  ( ( ph  \/  ps )  /\  -.  ps ) )  <->  ( ( ps  /\  -.  ph )  \/  ( ph  /\  -.  ps ) ) )
101, 2, 93bitri 199 . 2  |-  ( (
ph  \/_  ps )  <->  ( ( ps  /\  -.  ph )  \/  ( ph  /\ 
-.  ps ) ) )
11 orcom 657 . 2  |-  ( ( ( ps  /\  -.  ph )  \/  ( ph  /\ 
-.  ps ) )  <->  ( ( ph  /\  -.  ps )  \/  ( ps  /\  -.  ph ) ) )
12 ancom 257 . . 3  |-  ( ( ps  /\  -.  ph ) 
<->  ( -.  ph  /\  ps ) )
1312orbi2i 689 . 2  |-  ( ( ( ph  /\  -.  ps )  \/  ( ps  /\  -.  ph )
)  <->  ( ( ph  /\ 
-.  ps )  \/  ( -.  ph  /\  ps )
) )
1410, 11, 133bitri 199 1  |-  ( (
ph  \/_  ps )  <->  ( ( ph  /\  -.  ps )  \/  ( -.  ph  /\  ps )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 101    <-> wb 102    \/ wo 639    \/_ wxo 1282
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-xor 1283
This theorem is referenced by:  xordc  1299  symdifxor  3231
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