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Theorem xor3dc 1294
Description: Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 12-Apr-2018.)
Assertion
Ref Expression
xor3dc  |-  (DECID  ph  ->  (DECID  ps 
->  ( -.  ( ph  <->  ps )  <->  ( ph  <->  -.  ps )
) ) )

Proof of Theorem xor3dc
StepHypRef Expression
1 dcn 757 . . . . . 6  |-  (DECID  ps  -> DECID  -.  ps )
2 dcbi 855 . . . . . 6  |-  (DECID  ph  ->  (DECID  -. 
ps  -> DECID 
( ph  <->  -.  ps )
) )
31, 2syl5 32 . . . . 5  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( ph  <->  -.  ps )
) )
43imp 119 . . . 4  |-  ( (DECID  ph  /\ DECID  ps )  -> DECID 
( ph  <->  -.  ps )
)
5 pm5.18dc 788 . . . . . . 7  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( ph  <->  ps )  <->  -.  ( ph  <->  -.  ps )
) ) )
65imp 119 . . . . . 6  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( ph  <->  ps )  <->  -.  ( ph  <->  -. 
ps ) ) )
76a1d 22 . . . . 5  |-  ( (DECID  ph  /\ DECID  ps )  ->  (DECID  ( ph  <->  -.  ps )  ->  ( ( ph  <->  ps )  <->  -.  ( ph  <->  -.  ps )
) ) )
87con2biddc 785 . . . 4  |-  ( (DECID  ph  /\ DECID  ps )  ->  (DECID  ( ph  <->  -.  ps )  ->  ( ( ph  <->  -.  ps )  <->  -.  ( ph  <->  ps )
) ) )
94, 8mpd 13 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( ph  <->  -. 
ps )  <->  -.  ( ph 
<->  ps ) ) )
109bicomd 133 . 2  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( -.  ( ph 
<->  ps )  <->  ( ph  <->  -. 
ps ) ) )
1110ex 112 1  |-  (DECID  ph  ->  (DECID  ps 
->  ( -.  ( ph  <->  ps )  <->  ( ph  <->  -.  ps )
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    <-> wb 102  DECID wdc 753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-dc 754
This theorem is referenced by:  pm5.15dc  1296  xor2dc  1297  nbbndc  1301
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