Theorem bj-nnbist 13779

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 13792). (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 826	. . . 4 |- (STAB ph <-> ( -. -. ph -> ph ) )
2	1	biimpi 119	. . 3 |- (STAB ph -> ( -. -. ph -> ph ) )
3	2	com12 30	. 2 |- ( -. -. ph -> (STAB ph -> ph ) )
4		bj-trst 13774	. 2 |- ( ph -> STAB ph )
5	3, 4	impbid1 141	1 |- ( -. -. ph -> (STAB ph <-> ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104  STAB wstab 825

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 826

This theorem is referenced by:  bj-stst  13780  bj-nnbidc  13792  bj-stdc  13795