Theorem stdcndc 846

Description: A formula is decidable if and only if its negation is decidable and it is stable (that is, it is testable and stable). (Contributed by David A. Wheeler, 13-Aug-2018.) (Proof shortened by BJ, 28-Oct-2023.)

Assertion

Ref	Expression
stdcndc	⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) ↔ DECID 𝜑)

Proof of Theorem stdcndc

Step	Hyp	Ref	Expression
1		df-stab 832	. . . 4 ⊢ (STAB 𝜑 ↔ (¬ ¬ 𝜑 → 𝜑))
2		df-dc 836	. . . 4 ⊢ (DECID ¬ 𝜑 ↔ (¬ 𝜑 ∨ ¬ ¬ 𝜑))
3		pm2.36 805	. . . . 5 ⊢ ((¬ ¬ 𝜑 → 𝜑) → ((¬ 𝜑 ∨ ¬ ¬ 𝜑) → (𝜑 ∨ ¬ 𝜑)))
4	3	imp 124	. . . 4 ⊢ (((¬ ¬ 𝜑 → 𝜑) ∧ (¬ 𝜑 ∨ ¬ ¬ 𝜑)) → (𝜑 ∨ ¬ 𝜑))
5	1, 2, 4	syl2anb 291	. . 3 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) → (𝜑 ∨ ¬ 𝜑))
6		df-dc 836	. . 3 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
7	5, 6	sylibr 134	. 2 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) → DECID 𝜑)
8		dcstab 845	. . 3 ⊢ (DECID 𝜑 → STAB 𝜑)
9		dcn 843	. . 3 ⊢ (DECID 𝜑 → DECID ¬ 𝜑)
10	8, 9	jca 306	. 2 ⊢ (DECID 𝜑 → (STAB 𝜑 ∧ DECID ¬ 𝜑))
11	7, 10	impbii 126	1 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) ↔ DECID 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ 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-in1 615  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:  stdcn  848