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Theorem xor 902
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 426 . . . 4  |-  ( (
ph  ->  ps )  <->  -.  ( ph  /\  -.  ps )
)
2 iman 426 . . . 4  |-  ( ( ps  ->  ph )  <->  -.  ( ps  /\  -.  ph )
)
31, 2anbi12i 703 . . 3  |-  ( ( ( ph  ->  ps )  /\  ( ps  ->  ph ) )  <->  ( -.  ( ph  /\  -.  ps )  /\  -.  ( ps 
/\  -.  ph ) ) )
4 dfbi2 634 . . 3  |-  ( (
ph 
<->  ps )  <->  ( ( ph  ->  ps )  /\  ( ps  ->  ph )
) )
5 ioran 493 . . 3  |-  ( -.  ( ( ph  /\  -.  ps )  \/  ( ps  /\  -.  ph )
)  <->  ( -.  ( ph  /\  -.  ps )  /\  -.  ( ps  /\  -.  ph ) ) )
63, 4, 53bitr4ri 282 . 2  |-  ( -.  ( ( ph  /\  -.  ps )  \/  ( ps  /\  -.  ph )
)  <->  ( ph  <->  ps )
)
76con1bii 333 1  |-  ( -.  ( ph  <->  ps )  <->  ( ( ph  /\  -.  ps )  \/  ( ps  /\  -.  ph )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373
This theorem is referenced by:  dfbi3  904  pm5.24  905  4exmid  950  excxor  1411  elsymdif  3668  symdif2  3671  rpnnen2  14278  ist0-3  20361  prtlem90  32431  abnotataxb  38504  ldepslinc  40355
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