Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-nnbidc Unicode version

Theorem bj-nnbidc 13106
Description: If a formula is not refutable, then it is decidable if and only if it is provable. See also comment of bj-nnbist 13097. (Contributed by BJ, 24-Nov-2023.)
Assertion
Ref Expression
bj-nnbidc  |-  ( -. 
-.  ph  ->  (DECID  ph  <->  ph ) )

Proof of Theorem bj-nnbidc
StepHypRef Expression
1 bj-dcstab 13105 . . 3  |-  (DECID  ph  -> STAB  ph )
2 bj-nnbist 13097 . . 3  |-  ( -. 
-.  ph  ->  (STAB  ph  <->  ph ) )
31, 2syl5ib 153 . 2  |-  ( -. 
-.  ph  ->  (DECID  ph  ->  ph ) )
4 bj-trdc 13103 . 2  |-  ( ph  -> DECID  ph )
53, 4impbid1 141 1  |-  ( -. 
-.  ph  ->  (DECID  ph  <->  ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104  STAB wstab 815  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-in2 604  ax-io 698
This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820
This theorem is referenced by:  bj-dcdc  13109  bj-dcst  13111
  Copyright terms: Public domain W3C validator