Theorem bj-nnbist 15400

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 15413). (Contributed by BJ, 24-Nov-2023.)

Assertion

Ref	Expression
bj-nnbist	|- ( -. -. ph -> (STAB ph <-> ph ) )

Proof of Theorem bj-nnbist

Step	Hyp	Ref	Expression
1		df-stab 832	. . . 4 |- (STAB ph <-> ( -. -. ph -> ph ) )
2	1	biimpi 120	. . 3 |- (STAB ph -> ( -. -. ph -> ph ) )
3	2	com12 30	. 2 |- ( -. -. ph -> (STAB ph -> ph ) )
4		bj-trst 15395	. 2 |- ( ph -> STAB ph )
5	3, 4	impbid1 142	1 |- ( -. -. ph -> (STAB ph <-> ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105  STAB wstab 831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-stab 832

This theorem is referenced by:  bj-stst  15401  bj-nnbidc  15413  bj-stdc  15416