Theorem bj-dcstab 16457

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 843	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		bj-trst 16440	. . 3 ⊢ (𝜑 → STAB 𝜑)
3		bj-fast 16442	. . 3 ⊢ (¬ 𝜑 → STAB 𝜑)
4	2, 3	jaoi 724	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → STAB 𝜑)
5	1, 4	sylbi 121	1 ⊢ (DECID 𝜑 → STAB 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 716  STAB wstab 838  DECID wdc 842

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 620  ax-io 717

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843

This theorem is referenced by:  bj-nnbidc  16458