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Theorem dcim 818
Description: An implication between two decidable propositions is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.)
Assertion
Ref Expression
dcim  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( ph  ->  ps )
) )

Proof of Theorem dcim
StepHypRef Expression
1 df-dc 777 . 2  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
2 df-dc 777 . . . . . . . 8  |-  (DECID  ps  <->  ( ps  \/  -.  ps ) )
32anbi2i 445 . . . . . . 7  |-  ( (
ph  /\ DECID  ps )  <->  ( ph  /\  ( ps  \/  -.  ps ) ) )
4 andi 765 . . . . . . 7  |-  ( (
ph  /\  ( ps  \/  -.  ps ) )  <-> 
( ( ph  /\  ps )  \/  ( ph  /\  -.  ps )
) )
53, 4bitri 182 . . . . . 6  |-  ( (
ph  /\ DECID  ps )  <->  ( ( ph  /\  ps )  \/  ( ph  /\  -.  ps ) ) )
6 pm3.4 326 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  ( ph  ->  ps ) )
7 annimim 816 . . . . . . 7  |-  ( (
ph  /\  -.  ps )  ->  -.  ( ph  ->  ps ) )
86, 7orim12i 709 . . . . . 6  |-  ( ( ( ph  /\  ps )  \/  ( ph  /\ 
-.  ps ) )  -> 
( ( ph  ->  ps )  \/  -.  ( ph  ->  ps ) ) )
95, 8sylbi 119 . . . . 5  |-  ( (
ph  /\ DECID  ps )  ->  (
( ph  ->  ps )  \/  -.  ( ph  ->  ps ) ) )
10 df-dc 777 . . . . 5  |-  (DECID  ( ph  ->  ps )  <->  ( ( ph  ->  ps )  \/ 
-.  ( ph  ->  ps ) ) )
119, 10sylibr 132 . . . 4  |-  ( (
ph  /\ DECID  ps )  -> DECID  ( ph  ->  ps ) )
1211ex 113 . . 3  |-  ( ph  ->  (DECID  ps  -> DECID  ( ph  ->  ps ) ) )
13 ax-in2 578 . . . . 5  |-  ( -. 
ph  ->  ( ph  ->  ps ) )
1413a1d 22 . . . 4  |-  ( -. 
ph  ->  (DECID  ps  ->  ( ph  ->  ps ) ) )
15 orc 666 . . . . 5  |-  ( (
ph  ->  ps )  -> 
( ( ph  ->  ps )  \/  -.  ( ph  ->  ps ) ) )
1615, 10sylibr 132 . . . 4  |-  ( (
ph  ->  ps )  -> DECID  ( ph  ->  ps ) )
1714, 16syl6 33 . . 3  |-  ( -. 
ph  ->  (DECID  ps  -> DECID  ( ph  ->  ps ) ) )
1812, 17jaoi 669 . 2  |-  ( (
ph  \/  -.  ph )  ->  (DECID  ps  -> DECID  ( ph  ->  ps ) ) )
191, 18sylbi 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:  pm4.79dc  843  pm5.11dc  849  dcbi  878  annimdc  879
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