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Theorem pm5.17dc 844
Description: Two ways of stating exclusive-or which are equivalent for a decidable proposition. Based on theorem *5.17 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 16-Apr-2018.)
Assertion
Ref Expression
pm5.17dc  |-  (DECID  ps  ->  ( ( ( ph  \/  ps )  /\  -.  ( ph  /\  ps ) )  <-> 
( ph  <->  -.  ps )
) )

Proof of Theorem pm5.17dc
StepHypRef Expression
1 bicom 138 . 2  |-  ( (
ph 
<->  -.  ps )  <->  ( -.  ps 
<-> 
ph ) )
2 dfbi2 380 . . 3  |-  ( ( -.  ps  <->  ph )  <->  ( ( -.  ps  ->  ph )  /\  ( ph  ->  -.  ps )
) )
3 orcom 680 . . . . 5  |-  ( (
ph  \/  ps )  <->  ( ps  \/  ph )
)
4 dfordc 825 . . . . 5  |-  (DECID  ps  ->  ( ( ps  \/  ph ) 
<->  ( -.  ps  ->  ph ) ) )
53, 4syl5rbb 191 . . . 4  |-  (DECID  ps  ->  ( ( -.  ps  ->  ph )  <->  ( ph  \/  ps ) ) )
6 imnan 657 . . . . 5  |-  ( (
ph  ->  -.  ps )  <->  -.  ( ph  /\  ps ) )
76a1i 9 . . . 4  |-  (DECID  ps  ->  ( ( ph  ->  -.  ps )  <->  -.  ( ph  /\ 
ps ) ) )
85, 7anbi12d 457 . . 3  |-  (DECID  ps  ->  ( ( ( -.  ps  ->  ph )  /\  ( ph  ->  -.  ps )
)  <->  ( ( ph  \/  ps )  /\  -.  ( ph  /\  ps )
) ) )
92, 8syl5bb 190 . 2  |-  (DECID  ps  ->  ( ( -.  ps  <->  ph )  <->  ( ( ph  \/  ps )  /\  -.  ( ph  /\  ps ) ) ) )
101, 9syl5rbb 191 1  |-  (DECID  ps  ->  ( ( ( ph  \/  ps )  /\  -.  ( ph  /\  ps ) )  <-> 
( ph  <->  -.  ps )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662  DECID wdc 776
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 777
This theorem is referenced by:  xor2dc  1322
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