ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  stabtestimpdc Unicode version

Theorem stabtestimpdc 835
Description: "Stable and testable" is equivalent to decidable. (Contributed by David A. Wheeler, 13-Aug-2018.)
Assertion
Ref Expression
stabtestimpdc  |-  ( (STAB  ph  /\ DECID  -.  ph )  <-> DECID  ph )

Proof of Theorem stabtestimpdc
StepHypRef Expression
1 exmiddc 755 . . . . . 6  |-  (DECID  -.  ph  ->  ( -.  ph  \/  -.  -.  ph ) )
21adantl 266 . . . . 5  |-  ( (STAB  ph  /\ DECID  -.  ph )  ->  ( -.  ph  \/  -.  -.  ph ) )
3 df-stab 751 . . . . . . . 8  |-  (STAB  ph  <->  ( -.  -.  ph  ->  ph ) )
43biimpi 117 . . . . . . 7  |-  (STAB  ph  ->  ( -.  -.  ph  ->  ph ) )
54orim2d 712 . . . . . 6  |-  (STAB  ph  ->  ( ( -.  ph  \/  -.  -.  ph )  -> 
( -.  ph  \/  ph ) ) )
65adantr 265 . . . . 5  |-  ( (STAB  ph  /\ DECID  -.  ph )  ->  ( ( -.  ph  \/  -.  -.  ph )  ->  ( -.  ph  \/  ph ) ) )
72, 6mpd 13 . . . 4  |-  ( (STAB  ph  /\ DECID  -.  ph )  ->  ( -.  ph  \/  ph ) )
87orcomd 658 . . 3  |-  ( (STAB  ph  /\ DECID  -.  ph )  ->  ( ph  \/  -.  ph ) )
9 df-dc 754 . . 3  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
108, 9sylibr 141 . 2  |-  ( (STAB  ph  /\ DECID  -.  ph )  -> DECID  ph )
11 dcimpstab 763 . . 3  |-  (DECID  ph  -> STAB  ph )
12 dcn 757 . . 3  |-  (DECID  ph  -> DECID  -.  ph )
1311, 12jca 294 . 2  |-  (DECID  ph  ->  (STAB  ph  /\ DECID  -.  ph ) )
1410, 13impbii 121 1  |-  ( (STAB  ph  /\ DECID  -.  ph )  <-> DECID  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    <-> wb 102    \/ wo 639  STAB wstab 750  DECID wdc 753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-stab 751  df-dc 754
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator