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Theorem xor 861
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 413 . . . 4  |-  ( (
ph  ->  ps )  <->  -.  ( ph  /\  -.  ps )
)
2 iman 413 . . . 4  |-  ( ( ps  ->  ph )  <->  -.  ( ps  /\  -.  ph )
)
31, 2anbi12i 678 . . 3  |-  ( ( ( ph  ->  ps )  /\  ( ps  ->  ph ) )  <->  ( -.  ( ph  /\  -.  ps )  /\  -.  ( ps 
/\  -.  ph ) ) )
4 dfbi2 609 . . 3  |-  ( (
ph 
<->  ps )  <->  ( ( ph  ->  ps )  /\  ( ps  ->  ph )
) )
5 ioran 476 . . 3  |-  ( -.  ( ( ph  /\  -.  ps )  \/  ( ps  /\  -.  ph )
)  <->  ( -.  ( ph  /\  -.  ps )  /\  -.  ( ps  /\  -.  ph ) ) )
63, 4, 53bitr4ri 269 . 2  |-  ( -.  ( ( ph  /\  -.  ps )  \/  ( ps  /\  -.  ph )
)  <->  ( ph  <->  ps )
)
76con1bii 321 1  |-  ( -.  ( ph  <->  ps )  <->  ( ( ph  /\  -.  ps )  \/  ( ps  /\  -.  ph )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358
This theorem is referenced by:  dfbi3  863  pm5.24  864  4exmid  905  excxor  1300  symdif2  3434  rpnnen2  12504  ist0-3  17073  elsymdif  23778  prtlem90  26135  abnotataxb  27297
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360
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