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Theorem xor 866
Description: Two ways to express "exclusive or." Theorem *5.22 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 22-Jan-2013.)
Assertion
Ref Expression
xor  |-  ( -.  ( ph  <->  ps )  <->  ( ( ph  /\  -.  ps )  \/  ( ps  /\  -.  ph )
) )

Proof of Theorem xor
StepHypRef Expression
1 iman 415 . . . 4  |-  ( (
ph  ->  ps )  <->  -.  ( ph  /\  -.  ps )
)
2 iman 415 . . . 4  |-  ( ( ps  ->  ph )  <->  -.  ( ps  /\  -.  ph )
)
31, 2anbi12i 681 . . 3  |-  ( ( ( ph  ->  ps )  /\  ( ps  ->  ph ) )  <->  ( -.  ( ph  /\  -.  ps )  /\  -.  ( ps 
/\  -.  ph ) ) )
4 dfbi2 612 . . 3  |-  ( (
ph 
<->  ps )  <->  ( ( ph  ->  ps )  /\  ( ps  ->  ph )
) )
5 ioran 478 . . 3  |-  ( -.  ( ( ph  /\  -.  ps )  \/  ( ps  /\  -.  ph )
)  <->  ( -.  ( ph  /\  -.  ps )  /\  -.  ( ps  /\  -.  ph ) ) )
63, 4, 53bitr4ri 271 . 2  |-  ( -.  ( ( ph  /\  -.  ps )  \/  ( ps  /\  -.  ph )
)  <->  ( ph  <->  ps )
)
76con1bii 323 1  |-  ( -.  ( ph  <->  ps )  <->  ( ( ph  /\  -.  ps )  \/  ( ps  /\  -.  ph )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360
This theorem is referenced by:  dfbi3  868  pm5.24  869  4exmid  910  excxor  1305  symdif2  3341  rpnnen2  12378  ist0-3  16905  elsymdif  23542  prtlem90  25889  abnotataxb  26870
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362
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