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Theorem dcim 826
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 820 . 2  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
2 df-dc 820 . . . . . . . 8  |-  (DECID  ps  <->  ( ps  \/  -.  ps ) )
32anbi2i 452 . . . . . . 7  |-  ( (
ph  /\ DECID  ps )  <->  ( ph  /\  ( ps  \/  -.  ps ) ) )
4 andi 807 . . . . . . 7  |-  ( (
ph  /\  ( ps  \/  -.  ps ) )  <-> 
( ( ph  /\  ps )  \/  ( ph  /\  -.  ps )
) )
53, 4bitri 183 . . . . . 6  |-  ( (
ph  /\ DECID  ps )  <->  ( ( ph  /\  ps )  \/  ( ph  /\  -.  ps ) ) )
6 pm3.4 331 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  ( ph  ->  ps ) )
7 annimim 675 . . . . . . 7  |-  ( (
ph  /\  -.  ps )  ->  -.  ( ph  ->  ps ) )
86, 7orim12i 748 . . . . . 6  |-  ( ( ( ph  /\  ps )  \/  ( ph  /\ 
-.  ps ) )  -> 
( ( ph  ->  ps )  \/  -.  ( ph  ->  ps ) ) )
95, 8sylbi 120 . . . . 5  |-  ( (
ph  /\ DECID  ps )  ->  (
( ph  ->  ps )  \/  -.  ( ph  ->  ps ) ) )
10 df-dc 820 . . . . 5  |-  (DECID  ( ph  ->  ps )  <->  ( ( ph  ->  ps )  \/ 
-.  ( ph  ->  ps ) ) )
119, 10sylibr 133 . . . 4  |-  ( (
ph  /\ DECID  ps )  -> DECID  ( ph  ->  ps ) )
1211ex 114 . . 3  |-  ( ph  ->  (DECID  ps  -> DECID  ( ph  ->  ps ) ) )
13 ax-in2 604 . . . . 5  |-  ( -. 
ph  ->  ( ph  ->  ps ) )
1413a1d 22 . . . 4  |-  ( -. 
ph  ->  (DECID  ps  ->  ( ph  ->  ps ) ) )
15 orc 701 . . . . 5  |-  ( (
ph  ->  ps )  -> 
( ( ph  ->  ps )  \/  -.  ( ph  ->  ps ) ) )
1615, 10sylibr 133 . . . 4  |-  ( (
ph  ->  ps )  -> DECID  ( ph  ->  ps ) )
1714, 16syl6 33 . . 3  |-  ( -. 
ph  ->  (DECID  ps  -> DECID  ( ph  ->  ps ) ) )
1812, 17jaoi 705 . 2  |-  ( (
ph  \/  -.  ph )  ->  (DECID  ps  -> DECID  ( ph  ->  ps ) ) )
191, 18sylbi 120 1  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( ph  ->  ps )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 697  DECID wdc 819
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 603  ax-in2 604  ax-io 698
This theorem depends on definitions:  df-bi 116  df-dc 820
This theorem is referenced by:  pm4.79dc  888  pm5.11dc  894  dcbi  920  annimdc  921
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