Theorem stdcn 832

Description: A formula is stable if and only if the decidability of its negation implies its decidability. Note that the right-hand side of this biconditional is the converse of dcn 827. (Contributed by BJ, 18-Nov-2023.)

Assertion

Ref	Expression
stdcn	⊢ (STAB 𝜑 ↔ (DECID ¬ 𝜑 → DECID 𝜑))

Proof of Theorem stdcn

Step	Hyp	Ref	Expression
1		stdcndc 830	. . . 4 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) ↔ DECID 𝜑)
2	1	biimpi 119	. . 3 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) → DECID 𝜑)
3	2	ex 114	. 2 ⊢ (STAB 𝜑 → (DECID ¬ 𝜑 → DECID 𝜑))
4		olc 700	. . . . 5 ⊢ (¬ ¬ 𝜑 → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
5	4	imim1i 60	. . . 4 ⊢ (((¬ 𝜑 ∨ ¬ ¬ 𝜑) → (𝜑 ∨ ¬ 𝜑)) → (¬ ¬ 𝜑 → (𝜑 ∨ ¬ 𝜑)))
6		orel2 715	. . . 4 ⊢ (¬ ¬ 𝜑 → ((𝜑 ∨ ¬ 𝜑) → 𝜑))
7	5, 6	sylcom 28	. . 3 ⊢ (((¬ 𝜑 ∨ ¬ ¬ 𝜑) → (𝜑 ∨ ¬ 𝜑)) → (¬ ¬ 𝜑 → 𝜑))
8		df-dc 820	. . . 4 ⊢ (DECID ¬ 𝜑 ↔ (¬ 𝜑 ∨ ¬ ¬ 𝜑))
9		df-dc 820	. . . 4 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
10	8, 9	imbi12i 238	. . 3 ⊢ ((DECID ¬ 𝜑 → DECID 𝜑) ↔ ((¬ 𝜑 ∨ ¬ ¬ 𝜑) → (𝜑 ∨ ¬ 𝜑)))
11		df-stab 816	. . 3 ⊢ (STAB 𝜑 ↔ (¬ ¬ 𝜑 → 𝜑))
12	7, 10, 11	3imtr4i 200	. 2 ⊢ ((DECID ¬ 𝜑 → DECID 𝜑) → STAB 𝜑)
13	3, 12	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:  dcnn  833