Theorem bj-nnbidc 13638

Description: If a formula is not refutable, then it is decidable if and only if it is provable. See also comment of bj-nnbist 13625. (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 13637	. . 3 ⊢ (DECID 𝜑 → STAB 𝜑)
2		bj-nnbist 13625	. . 3 ⊢ (¬ ¬ 𝜑 → (STAB 𝜑 ↔ 𝜑))
3	1, 2	syl5ib 153	. 2 ⊢ (¬ ¬ 𝜑 → (DECID 𝜑 → 𝜑))
4		bj-trdc 13633	. 2 ⊢ (𝜑 → DECID 𝜑)
5	3, 4	impbid1 141	1 ⊢ (¬ ¬ 𝜑 → (DECID 𝜑 ↔ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104  STAB wstab 820  DECID wdc 824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825

This theorem is referenced by:  bj-dcdc  13640  bj-dcst  13642