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Theorem dcan 876
Description: A conjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.)
Assertion
Ref Expression
dcan  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( ph  /\  ps )
) )

Proof of Theorem dcan
StepHypRef Expression
1 simpl 107 . . . . . 6  |-  ( ( -.  ph  /\  ps )  ->  -.  ph )
21intnanrd 875 . . . . 5  |-  ( ( -.  ph  /\  ps )  ->  -.  ( ph  /\  ps ) )
32orim2i 711 . . . 4  |-  ( ( ( ph  /\  ps )  \/  ( -.  ph 
/\  ps ) )  -> 
( ( ph  /\  ps )  \/  -.  ( ph  /\  ps )
) )
4 simpr 108 . . . . . 6  |-  ( ( ( ph  \/  -.  ph )  /\  -.  ps )  ->  -.  ps )
54intnand 874 . . . . 5  |-  ( ( ( ph  \/  -.  ph )  /\  -.  ps )  ->  -.  ( ph  /\ 
ps ) )
65olcd 686 . . . 4  |-  ( ( ( ph  \/  -.  ph )  /\  -.  ps )  ->  ( ( ph  /\ 
ps )  \/  -.  ( ph  /\  ps )
) )
73, 6jaoi 669 . . 3  |-  ( ( ( ( ph  /\  ps )  \/  ( -.  ph  /\  ps )
)  \/  ( (
ph  \/  -.  ph )  /\  -.  ps ) )  ->  ( ( ph  /\ 
ps )  \/  -.  ( ph  /\  ps )
) )
8 df-dc 777 . . . . 5  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
9 df-dc 777 . . . . 5  |-  (DECID  ps  <->  ( ps  \/  -.  ps ) )
108, 9anbi12i 448 . . . 4  |-  ( (DECID  ph  /\ DECID  ps ) 
<->  ( ( ph  \/  -.  ph )  /\  ( ps  \/  -.  ps )
) )
11 andi 765 . . . 4  |-  ( ( ( ph  \/  -.  ph )  /\  ( ps  \/  -.  ps )
)  <->  ( ( (
ph  \/  -.  ph )  /\  ps )  \/  (
( ph  \/  -.  ph )  /\  -.  ps ) ) )
12 andir 766 . . . . 5  |-  ( ( ( ph  \/  -.  ph )  /\  ps )  <->  ( ( ph  /\  ps )  \/  ( -.  ph 
/\  ps ) ) )
1312orbi1i 713 . . . 4  |-  ( ( ( ( ph  \/  -.  ph )  /\  ps )  \/  ( ( ph  \/  -.  ph )  /\  -.  ps ) )  <-> 
( ( ( ph  /\ 
ps )  \/  ( -.  ph  /\  ps )
)  \/  ( (
ph  \/  -.  ph )  /\  -.  ps ) ) )
1410, 11, 133bitri 204 . . 3  |-  ( (DECID  ph  /\ DECID  ps ) 
<->  ( ( ( ph  /\ 
ps )  \/  ( -.  ph  /\  ps )
)  \/  ( (
ph  \/  -.  ph )  /\  -.  ps ) ) )
15 df-dc 777 . . 3  |-  (DECID  ( ph  /\ 
ps )  <->  ( ( ph  /\  ps )  \/ 
-.  ( ph  /\  ps ) ) )
167, 14, 153imtr4i 199 . 2  |-  ( (DECID  ph  /\ DECID  ps )  -> DECID 
( ph  /\  ps )
)
1716ex 113 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:  dcbi  878  annimdc  879  pm4.55dc  880  orandc  881  anordc  898  xordidc  1331  nn0n0n1ge2b  8497  gcdmndc  10473  gcdsupex  10482  gcdsupcl  10483  gcdaddm  10508  lcmval  10578  lcmcllem  10582  lcmledvds  10585
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