Theorem stabtestimpdc 857

Description: "Stable and testable" is equivalent to decidable. (Contributed by David A. Wheeler, 13-Aug-2018.)

Assertion

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

Proof of Theorem stabtestimpdc

Step	Hyp	Ref	Expression
1		exmiddc 777	. . . . . 6 ⊢ (DECID ¬ 𝜑 → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
2	1	adantl 271	. . . . 5 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
3		df-stab 773	. . . . . . . 8 ⊢ (STAB 𝜑 ↔ (¬ ¬ 𝜑 → 𝜑))
4	3	biimpi 118	. . . . . . 7 ⊢ (STAB 𝜑 → (¬ ¬ 𝜑 → 𝜑))
5	4	orim2d 734	. . . . . 6 ⊢ (STAB 𝜑 → ((¬ 𝜑 ∨ ¬ ¬ 𝜑) → (¬ 𝜑 ∨ 𝜑)))
6	5	adantr 270	. . . . 5 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) → ((¬ 𝜑 ∨ ¬ ¬ 𝜑) → (¬ 𝜑 ∨ 𝜑)))
7	2, 6	mpd 13	. . . 4 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) → (¬ 𝜑 ∨ 𝜑))
8	7	orcomd 680	. . 3 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) → (𝜑 ∨ ¬ 𝜑))
9		df-dc 776	. . 3 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
10	8, 9	sylibr 132	. 2 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) → DECID 𝜑)
11		dcimpstab 785	. . 3 ⊢ (DECID 𝜑 → STAB 𝜑)
12		dcn 779	. . 3 ⊢ (DECID 𝜑 → DECID ¬ 𝜑)
13	11, 12	jca 300	. 2 ⊢ (DECID 𝜑 → (STAB 𝜑 ∧ DECID ¬ 𝜑))
14	10, 13	impbii 124	1 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) ↔ DECID 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 661  STAB wstab 772  DECID wdc 775

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-stab 773  df-dc 776

This theorem is referenced by: (None)