Theorem bj-nnbist 15390

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 15403). (Contributed by BJ, 24-Nov-2023.)

Assertion

Ref	Expression
bj-nnbist	⊢ (¬ ¬ 𝜑 → (STAB 𝜑 ↔ 𝜑))

Proof of Theorem bj-nnbist

Step	Hyp	Ref	Expression
1		df-stab 832	. . . 4 ⊢ (STAB 𝜑 ↔ (¬ ¬ 𝜑 → 𝜑))
2	1	biimpi 120	. . 3 ⊢ (STAB 𝜑 → (¬ ¬ 𝜑 → 𝜑))
3	2	com12 30	. 2 ⊢ (¬ ¬ 𝜑 → (STAB 𝜑 → 𝜑))
4		bj-trst 15385	. 2 ⊢ (𝜑 → STAB 𝜑)
5	3, 4	impbid1 142	1 ⊢ (¬ ¬ 𝜑 → (STAB 𝜑 ↔ 𝜑))

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  15391  bj-nnbidc  15403  bj-stdc  15406