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Theorem xordc 1370
Description: Two ways to express "exclusive or" between decidable propositions. Theorem *5.22 of [WhiteheadRussell] p. 124, but for decidable propositions. (Contributed by Jim Kingdon, 5-May-2018.)
Assertion
Ref Expression
xordc  |-  (DECID  ph  ->  (DECID  ps 
->  ( -.  ( ph  <->  ps )  <->  ( ( ph  /\ 
-.  ps )  \/  ( ps  /\  -.  ph )
) ) ) )

Proof of Theorem xordc
StepHypRef Expression
1 xornbidc 1369 . . . 4  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( ph  \/_  ps ) 
<->  -.  ( ph  <->  ps )
) ) )
21imp 123 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( ph  \/_ 
ps )  <->  -.  ( ph 
<->  ps ) ) )
3 excxor 1356 . . . 4  |-  ( (
ph  \/_  ps )  <->  ( ( ph  /\  -.  ps )  \/  ( -.  ph  /\  ps )
) )
4 ancom 264 . . . . 5  |-  ( ( -.  ph  /\  ps )  <->  ( ps  /\  -.  ph ) )
54orbi2i 751 . . . 4  |-  ( ( ( ph  /\  -.  ps )  \/  ( -.  ph  /\  ps )
)  <->  ( ( ph  /\ 
-.  ps )  \/  ( ps  /\  -.  ph )
) )
63, 5bitri 183 . . 3  |-  ( (
ph  \/_  ps )  <->  ( ( ph  /\  -.  ps )  \/  ( ps  /\  -.  ph )
) )
72, 6bitr3di 194 . 2  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( -.  ( ph 
<->  ps )  <->  ( ( ph  /\  -.  ps )  \/  ( ps  /\  -.  ph ) ) ) )
87ex 114 1  |-  (DECID  ph  ->  (DECID  ps 
->  ( -.  ( ph  <->  ps )  <->  ( ( ph  /\ 
-.  ps )  \/  ( ps  /\  -.  ph )
) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697  DECID wdc 819    \/_ wxo 1353
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 603  ax-in2 604  ax-io 698
This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-xor 1354
This theorem is referenced by:  dfbi3dc  1375  pm5.24dc  1376
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