Theorem stdcndc 830

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 816	. . . 4 ⊢ (STAB 𝜑 ↔ (¬ ¬ 𝜑 → 𝜑))
2		df-dc 820	. . . 4 ⊢ (DECID ¬ 𝜑 ↔ (¬ 𝜑 ∨ ¬ ¬ 𝜑))
3		pm2.36 793	. . . . 5 ⊢ ((¬ ¬ 𝜑 → 𝜑) → ((¬ 𝜑 ∨ ¬ ¬ 𝜑) → (𝜑 ∨ ¬ 𝜑)))
4	3	imp 123	. . . 4 ⊢ (((¬ ¬ 𝜑 → 𝜑) ∧ (¬ 𝜑 ∨ ¬ ¬ 𝜑)) → (𝜑 ∨ ¬ 𝜑))
5	1, 2, 4	syl2anb 289	. . 3 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) → (𝜑 ∨ ¬ 𝜑))
6		df-dc 820	. . 3 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
7	5, 6	sylibr 133	. 2 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) → DECID 𝜑)
8		dcstab 829	. . 3 ⊢ (DECID 𝜑 → STAB 𝜑)
9		dcn 827	. . 3 ⊢ (DECID 𝜑 → DECID ¬ 𝜑)
10	8, 9	jca 304	. 2 ⊢ (DECID 𝜑 → (STAB 𝜑 ∧ DECID ¬ 𝜑))
11	7, 10	impbii 125	1 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) ↔ DECID 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697  STAB wstab 815  DECID wdc 819

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-in1 603  ax-in2 604  ax-io 698

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820

This theorem is referenced by:  stdcn  832