Theorem bj-dcstab 15402

Description: A decidable formula is stable. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-dcstab	⊢ (DECID 𝜑 → STAB 𝜑)

Proof of Theorem bj-dcstab

Step	Hyp	Ref	Expression
1		df-dc 836	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		bj-trst 15385	. . 3 ⊢ (𝜑 → STAB 𝜑)
3		bj-fast 15387	. . 3 ⊢ (¬ 𝜑 → STAB 𝜑)
4	2, 3	jaoi 717	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → STAB 𝜑)
5	1, 4	sylbi 121	1 ⊢ (DECID 𝜑 → STAB 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 709  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-nnbidc  15403