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Theorem bj-nnbist 13006
Description: If a formula is not refutable, then it is stable if and only if it is provable. By double-negation translation, if 𝜑 is a classical tautology, then ¬ ¬ 𝜑 is an intuitionistic tautology. Therefore, if 𝜑 is a classical tautology, then 𝜑 is intuitionistically equivalent to its stability (and to its decidability, see bj-nnbidc 13015). (Contributed by BJ, 24-Nov-2023.)
Assertion
Ref Expression
bj-nnbist (¬ ¬ 𝜑 → (STAB 𝜑𝜑))

Proof of Theorem bj-nnbist
StepHypRef Expression
1 df-stab 816 . . . 4 (STAB 𝜑 ↔ (¬ ¬ 𝜑𝜑))
21biimpi 119 . . 3 (STAB 𝜑 → (¬ ¬ 𝜑𝜑))
32com12 30 . 2 (¬ ¬ 𝜑 → (STAB 𝜑𝜑))
4 bj-trst 13004 . 2 (𝜑STAB 𝜑)
53, 4impbid1 141 1 (¬ ¬ 𝜑 → (STAB 𝜑𝜑))
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  13015  bj-stdc  13019  bj-stst  13021
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