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Theorem xornbidc 1298
Description: Exclusive or is equivalent to negated biconditional for decidable propositions. (Contributed by Jim Kingdon, 27-Apr-2018.)
Assertion
Ref Expression
xornbidc  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( ph  \/_  ps ) 
<->  -.  ( ph  <->  ps )
) ) )

Proof of Theorem xornbidc
StepHypRef Expression
1 xor2dc 1297 . . . 4  |-  (DECID  ph  ->  (DECID  ps 
->  ( -.  ( ph  <->  ps )  <->  ( ( ph  \/  ps )  /\  -.  ( ph  /\  ps )
) ) ) )
21imp 119 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( -.  ( ph 
<->  ps )  <->  ( ( ph  \/  ps )  /\  -.  ( ph  /\  ps ) ) ) )
3 df-xor 1283 . . 3  |-  ( (
ph  \/_  ps )  <->  ( ( ph  \/  ps )  /\  -.  ( ph  /\ 
ps ) ) )
42, 3syl6rbbr 192 . 2  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( ph  \/_ 
ps )  <->  -.  ( ph 
<->  ps ) ) )
54ex 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    \/ wo 639  DECID wdc 753    \/_ wxo 1282
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  df-xor 1283
This theorem is referenced by:  xordc  1299  xordidc  1306
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