Theorem bj-dcstab 13637

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 825	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		bj-trst 13620	. . 3 ⊢ (𝜑 → STAB 𝜑)
3		bj-fast 13622	. . 3 ⊢ (¬ 𝜑 → STAB 𝜑)
4	2, 3	jaoi 706	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → STAB 𝜑)
5	1, 4	sylbi 120	1 ⊢ (DECID 𝜑 → STAB 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 698  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-nnbidc  13638