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Theorem dcor 843
Description: A disjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 21-Apr-2018.)
Assertion
Ref Expression
dcor  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( ph  \/  ps )
) )

Proof of Theorem dcor
StepHypRef Expression
1 df-dc 743 . 2  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
2 orc 633 . . . . . 6  |-  ( ph  ->  ( ph  \/  ps ) )
32orcd 652 . . . . 5  |-  ( ph  ->  ( ( ph  \/  ps )  \/  -.  ( ph  \/  ps )
) )
4 df-dc 743 . . . . 5  |-  (DECID  ( ph  \/  ps )  <->  ( ( ph  \/  ps )  \/ 
-.  ( ph  \/  ps ) ) )
53, 4sylibr 137 . . . 4  |-  ( ph  -> DECID  (
ph  \/  ps )
)
65a1d 22 . . 3  |-  ( ph  ->  (DECID  ps  -> DECID  ( ph  \/  ps ) ) )
7 df-dc 743 . . . . 5  |-  (DECID  ps  <->  ( ps  \/  -.  ps ) )
8 olc 632 . . . . . . . . 9  |-  ( ps 
->  ( ph  \/  ps ) )
98adantl 262 . . . . . . . 8  |-  ( ( -.  ph  /\  ps )  ->  ( ph  \/  ps ) )
109orcd 652 . . . . . . 7  |-  ( ( -.  ph  /\  ps )  ->  ( ( ph  \/  ps )  \/  -.  ( ph  \/  ps )
) )
1110, 4sylibr 137 . . . . . 6  |-  ( ( -.  ph  /\  ps )  -> DECID  (
ph  \/  ps )
)
12 ioran 669 . . . . . . . . 9  |-  ( -.  ( ph  \/  ps ) 
<->  ( -.  ph  /\  -.  ps ) )
1312biimpri 124 . . . . . . . 8  |-  ( ( -.  ph  /\  -.  ps )  ->  -.  ( ph  \/  ps ) )
1413olcd 653 . . . . . . 7  |-  ( ( -.  ph  /\  -.  ps )  ->  ( ( ph  \/  ps )  \/  -.  ( ph  \/  ps )
) )
1514, 4sylibr 137 . . . . . 6  |-  ( ( -.  ph  /\  -.  ps )  -> DECID 
( ph  \/  ps ) )
1611, 15jaodan 710 . . . . 5  |-  ( ( -.  ph  /\  ( ps  \/  -.  ps )
)  -> DECID  ( ph  \/  ps ) )
177, 16sylan2b 271 . . . 4  |-  ( ( -.  ph  /\ DECID  ps )  -> DECID  ( ph  \/  ps ) )
1817ex 108 . . 3  |-  ( -. 
ph  ->  (DECID  ps  -> DECID  ( ph  \/  ps ) ) )
196, 18jaoi 636 . 2  |-  ( (
ph  \/  -.  ph )  ->  (DECID  ps  -> DECID  ( ph  \/  ps ) ) )
201, 19sylbi 114 1  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( ph  \/  ps )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 97    \/ wo 629  DECID wdc 742
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630
This theorem depends on definitions:  df-bi 110  df-dc 743
This theorem is referenced by:  pm4.55dc  846  pm3.12dc  865  pm3.13dc  866  dn1dc  867  eueq3dc  2715  distrlem4prl  6682  distrlem4pru  6683
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