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Theorem stdcndcOLD 814
 Description: Obsolete version of stdcndc 813 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 𝜑DECID ¬ 𝜑) ↔ DECID 𝜑)

Proof of Theorem stdcndcOLD
StepHypRef Expression
1 exmiddc 804 . . . . . 6 (DECID ¬ 𝜑 → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
21adantl 273 . . . . 5 ((STAB 𝜑DECID ¬ 𝜑) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
3 df-stab 799 . . . . . . . 8 (STAB 𝜑 ↔ (¬ ¬ 𝜑𝜑))
43biimpi 119 . . . . . . 7 (STAB 𝜑 → (¬ ¬ 𝜑𝜑))
54orim2d 760 . . . . . 6 (STAB 𝜑 → ((¬ 𝜑 ∨ ¬ ¬ 𝜑) → (¬ 𝜑𝜑)))
65adantr 272 . . . . 5 ((STAB 𝜑DECID ¬ 𝜑) → ((¬ 𝜑 ∨ ¬ ¬ 𝜑) → (¬ 𝜑𝜑)))
72, 6mpd 13 . . . 4 ((STAB 𝜑DECID ¬ 𝜑) → (¬ 𝜑𝜑))
87orcomd 701 . . 3 ((STAB 𝜑DECID ¬ 𝜑) → (𝜑 ∨ ¬ 𝜑))
9 df-dc 803 . . 3 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
108, 9sylibr 133 . 2 ((STAB 𝜑DECID ¬ 𝜑) → DECID 𝜑)
11 dcstab 812 . . 3 (DECID 𝜑STAB 𝜑)
12 dcn 810 . . 3 (DECID 𝜑DECID ¬ 𝜑)
1311, 12jca 302 . 2 (DECID 𝜑 → (STAB 𝜑DECID ¬ 𝜑))
1410, 13impbii 125 1 ((STAB 𝜑DECID ¬ 𝜑) ↔ DECID 𝜑)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 680  STAB wstab 798  DECID wdc 802 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 586  ax-in2 587  ax-io 681 This theorem depends on definitions:  df-bi 116  df-stab 799  df-dc 803 This theorem is referenced by: (None)
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