Theorem bj-nnbidc 15413

Description: If a formula is not refutable, then it is decidable if and only if it is provable. See also comment of bj-nnbist 15400. (Contributed by BJ, 24-Nov-2023.)

Assertion

Ref	Expression
bj-nnbidc	⊢ (¬ ¬ 𝜑 → (DECID 𝜑 ↔ 𝜑))

Proof of Theorem bj-nnbidc

Step	Hyp	Ref	Expression
1		bj-dcstab 15412	. . 3 ⊢ (DECID 𝜑 → STAB 𝜑)
2		bj-nnbist 15400	. . 3 ⊢ (¬ ¬ 𝜑 → (STAB 𝜑 ↔ 𝜑))
3	1, 2	imbitrid 154	. 2 ⊢ (¬ ¬ 𝜑 → (DECID 𝜑 → 𝜑))
4		bj-trdc 15408	. 2 ⊢ (𝜑 → DECID 𝜑)
5	3, 4	impbid1 142	1 ⊢ (¬ ¬ 𝜑 → (DECID 𝜑 ↔ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105  STAB wstab 831  DECID wdc 835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836

This theorem is referenced by:  bj-dcdc  15415  bj-dcst  15417