Theorem stdcndcOLD 851

Description: Obsolete version of stdcndc 850 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 841	. . . . . 6 ⊢ (DECID ¬ 𝜑 → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
2	1	adantl 277	. . . . 5 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
3		df-stab 836	. . . . . . . 8 ⊢ (STAB 𝜑 ↔ (¬ ¬ 𝜑 → 𝜑))
4	3	biimpi 120	. . . . . . 7 ⊢ (STAB 𝜑 → (¬ ¬ 𝜑 → 𝜑))
5	4	orim2d 793	. . . . . 6 ⊢ (STAB 𝜑 → ((¬ 𝜑 ∨ ¬ ¬ 𝜑) → (¬ 𝜑 ∨ 𝜑)))
6	5	adantr 276	. . . . 5 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) → ((¬ 𝜑 ∨ ¬ ¬ 𝜑) → (¬ 𝜑 ∨ 𝜑)))
7	2, 6	mpd 13	. . . 4 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) → (¬ 𝜑 ∨ 𝜑))
8	7	orcomd 734	. . 3 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) → (𝜑 ∨ ¬ 𝜑))
9		df-dc 840	. . 3 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
10	8, 9	sylibr 134	. 2 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) → DECID 𝜑)
11		dcstab 849	. . 3 ⊢ (DECID 𝜑 → STAB 𝜑)
12		dcn 847	. . 3 ⊢ (DECID 𝜑 → DECID ¬ 𝜑)
13	11, 12	jca 306	. 2 ⊢ (DECID 𝜑 → (STAB 𝜑 ∧ DECID ¬ 𝜑))
14	10, 13	impbii 126	1 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) ↔ DECID 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 713  STAB wstab 835  DECID wdc 839

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 617  ax-in2 618  ax-io 714

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840

This theorem is referenced by: (None)