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Theorem bj-nnbist 12938
Description: If a formula is not refutable, then it is stable if and only if it is provable. By double-negation translation, if  ph is a classical tautology, then  -.  -.  ph is an intuitionistic tautology. Therefore, if  ph is a classical tautology, then  ph is intuitionistically equivalent to its stability (and to its decidability, see bj-nnbidc 12947). (Contributed by BJ, 24-Nov-2023.)
Assertion
Ref Expression
bj-nnbist  |-  ( -. 
-.  ph  ->  (STAB  ph  <->  ph ) )

Proof of Theorem bj-nnbist
StepHypRef Expression
1 df-stab 816 . . . 4  |-  (STAB  ph  <->  ( -.  -.  ph  ->  ph ) )
21biimpi 119 . . 3  |-  (STAB  ph  ->  ( -.  -.  ph  ->  ph ) )
32com12 30 . 2  |-  ( -. 
-.  ph  ->  (STAB  ph  ->  ph ) )
4 bj-trst 12936 . 2  |-  ( ph  -> STAB  ph )
53, 4impbid1 141 1  |-  ( -. 
-.  ph  ->  (STAB  ph  <->  ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104  STAB wstab 815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107
This theorem depends on definitions:  df-bi 116  df-stab 816
This theorem is referenced by:  bj-nnbidc  12947  bj-stdc  12951  bj-stst  12953
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