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Theorem pm5.24dc 1377
Description: Theorem *5.24 of [WhiteheadRussell] p. 124, but for decidable propositions. (Contributed by Jim Kingdon, 5-May-2018.)
Assertion
Ref Expression
pm5.24dc  |-  (DECID  ph  ->  (DECID  ps 
->  ( -.  ( (
ph  /\  ps )  \/  ( -.  ph  /\  -.  ps ) )  <->  ( ( ph  /\  -.  ps )  \/  ( ps  /\  -.  ph ) ) ) ) )

Proof of Theorem pm5.24dc
StepHypRef Expression
1 dfbi3dc 1376 . . . . 5  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( ph  <->  ps )  <->  ( ( ph  /\  ps )  \/  ( -.  ph 
/\  -.  ps )
) ) ) )
21imp 123 . . . 4  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( ph  <->  ps )  <->  ( ( ph  /\ 
ps )  \/  ( -.  ph  /\  -.  ps ) ) ) )
32notbid 657 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( -.  ( ph 
<->  ps )  <->  -.  (
( ph  /\  ps )  \/  ( -.  ph  /\  -.  ps ) ) ) )
4 xordc 1371 . . . 4  |-  (DECID  ph  ->  (DECID  ps 
->  ( -.  ( ph  <->  ps )  <->  ( ( ph  /\ 
-.  ps )  \/  ( ps  /\  -.  ph )
) ) ) )
54imp 123 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( -.  ( ph 
<->  ps )  <->  ( ( ph  /\  -.  ps )  \/  ( ps  /\  -.  ph ) ) ) )
63, 5bitr3d 189 . 2  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( -.  (
( ph  /\  ps )  \/  ( -.  ph  /\  -.  ps ) )  <->  ( ( ph  /\  -.  ps )  \/  ( ps  /\  -.  ph ) ) ) )
76ex 114 1  |-  (DECID  ph  ->  (DECID  ps 
->  ( -.  ( (
ph  /\  ps )  \/  ( -.  ph  /\  -.  ps ) )  <->  ( ( ph  /\  -.  ps )  \/  ( ps  /\  -.  ph ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 698  DECID wdc 820
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 604  ax-in2 605  ax-io 699
This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-xor 1355
This theorem is referenced by: (None)
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