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Theorem dcor 877
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 777 . 2  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
2 orc 666 . . . . . 6  |-  ( ph  ->  ( ph  \/  ps ) )
32orcd 685 . . . . 5  |-  ( ph  ->  ( ( ph  \/  ps )  \/  -.  ( ph  \/  ps )
) )
4 df-dc 777 . . . . 5  |-  (DECID  ( ph  \/  ps )  <->  ( ( ph  \/  ps )  \/ 
-.  ( ph  \/  ps ) ) )
53, 4sylibr 132 . . . 4  |-  ( ph  -> DECID  (
ph  \/  ps )
)
65a1d 22 . . 3  |-  ( ph  ->  (DECID  ps  -> DECID  ( ph  \/  ps ) ) )
7 df-dc 777 . . . . 5  |-  (DECID  ps  <->  ( ps  \/  -.  ps ) )
8 olc 665 . . . . . . . . 9  |-  ( ps 
->  ( ph  \/  ps ) )
98adantl 271 . . . . . . . 8  |-  ( ( -.  ph  /\  ps )  ->  ( ph  \/  ps ) )
109orcd 685 . . . . . . 7  |-  ( ( -.  ph  /\  ps )  ->  ( ( ph  \/  ps )  \/  -.  ( ph  \/  ps )
) )
1110, 4sylibr 132 . . . . . 6  |-  ( ( -.  ph  /\  ps )  -> DECID  (
ph  \/  ps )
)
12 ioran 702 . . . . . . . . 9  |-  ( -.  ( ph  \/  ps ) 
<->  ( -.  ph  /\  -.  ps ) )
1312biimpri 131 . . . . . . . 8  |-  ( ( -.  ph  /\  -.  ps )  ->  -.  ( ph  \/  ps ) )
1413olcd 686 . . . . . . 7  |-  ( ( -.  ph  /\  -.  ps )  ->  ( ( ph  \/  ps )  \/  -.  ( ph  \/  ps )
) )
1514, 4sylibr 132 . . . . . 6  |-  ( ( -.  ph  /\  -.  ps )  -> DECID 
( ph  \/  ps ) )
1611, 15jaodan 744 . . . . 5  |-  ( ( -.  ph  /\  ( ps  \/  -.  ps )
)  -> DECID  ( ph  \/  ps ) )
177, 16sylan2b 281 . . . 4  |-  ( ( -.  ph  /\ DECID  ps )  -> DECID  ( ph  \/  ps ) )
1817ex 113 . . 3  |-  ( -. 
ph  ->  (DECID  ps  -> DECID  ( ph  \/  ps ) ) )
196, 18jaoi 669 . 2  |-  ( (
ph  \/  -.  ph )  ->  (DECID  ps  -> DECID  ( ph  \/  ps ) ) )
201, 19sylbi 119 1  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( ph  \/  ps )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    \/ 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:  pm4.55dc  880  orandc  881  pm3.12dc  900  pm3.13dc  901  dn1dc  902  eueq3dc  2767  distrlem4prl  6825  distrlem4pru  6826  exfzdc  9315  lcmmndc  10577  isprm3  10633
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