Theorem stdcndcOLD 831

Description: Obsolete version of stdcndc 830 as of 28-Oct-2023. (Contributed by David A. Wheeler, 13-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem stdcndcOLD

Step	Hyp	Ref	Expression
1		exmiddc 821	. . . . . 6 ⊢ (DECID ¬ 𝜑 → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
2	1	adantl 275	. . . . 5 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
3		df-stab 816	. . . . . . . 8 ⊢ (STAB 𝜑 ↔ (¬ ¬ 𝜑 → 𝜑))
4	3	biimpi 119	. . . . . . 7 ⊢ (STAB 𝜑 → (¬ ¬ 𝜑 → 𝜑))
5	4	orim2d 777	. . . . . 6 ⊢ (STAB 𝜑 → ((¬ 𝜑 ∨ ¬ ¬ 𝜑) → (¬ 𝜑 ∨ 𝜑)))
6	5	adantr 274	. . . . 5 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) → ((¬ 𝜑 ∨ ¬ ¬ 𝜑) → (¬ 𝜑 ∨ 𝜑)))
7	2, 6	mpd 13	. . . 4 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) → (¬ 𝜑 ∨ 𝜑))
8	7	orcomd 718	. . 3 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) → (𝜑 ∨ ¬ 𝜑))
9		df-dc 820	. . 3 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
10	8, 9	sylibr 133	. 2 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) → DECID 𝜑)
11		dcstab 829	. . 3 ⊢ (DECID 𝜑 → STAB 𝜑)
12		dcn 827	. . 3 ⊢ (DECID 𝜑 → DECID ¬ 𝜑)
13	11, 12	jca 304	. 2 ⊢ (DECID 𝜑 → (STAB 𝜑 ∧ DECID ¬ 𝜑))
14	10, 13	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: (None)