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Theorem stabtestimpdc 858
Description: "Stable and testable" is equivalent to decidable. (Contributed by David A. Wheeler, 13-Aug-2018.)
Assertion
Ref Expression
stabtestimpdc ((STAB 𝜑DECID ¬ 𝜑) ↔ DECID 𝜑)

Proof of Theorem stabtestimpdc
StepHypRef Expression
1 exmiddc 778 . . . . . 6 (DECID ¬ 𝜑 → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
21adantl 271 . . . . 5 ((STAB 𝜑DECID ¬ 𝜑) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
3 df-stab 774 . . . . . . . 8 (STAB 𝜑 ↔ (¬ ¬ 𝜑𝜑))
43biimpi 118 . . . . . . 7 (STAB 𝜑 → (¬ ¬ 𝜑𝜑))
54orim2d 735 . . . . . 6 (STAB 𝜑 → ((¬ 𝜑 ∨ ¬ ¬ 𝜑) → (¬ 𝜑𝜑)))
65adantr 270 . . . . 5 ((STAB 𝜑DECID ¬ 𝜑) → ((¬ 𝜑 ∨ ¬ ¬ 𝜑) → (¬ 𝜑𝜑)))
72, 6mpd 13 . . . 4 ((STAB 𝜑DECID ¬ 𝜑) → (¬ 𝜑𝜑))
87orcomd 681 . . 3 ((STAB 𝜑DECID ¬ 𝜑) → (𝜑 ∨ ¬ 𝜑))
9 df-dc 777 . . 3 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
108, 9sylibr 132 . 2 ((STAB 𝜑DECID ¬ 𝜑) → DECID 𝜑)
11 dcimpstab 786 . . 3 (DECID 𝜑STAB 𝜑)
12 dcn 780 . . 3 (DECID 𝜑DECID ¬ 𝜑)
1311, 12jca 300 . 2 (DECID 𝜑 → (STAB 𝜑DECID ¬ 𝜑))
1410, 13impbii 124 1 ((STAB 𝜑DECID ¬ 𝜑) ↔ DECID 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662  STAB wstab 773  DECID wdc 776
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 577  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-stab 774  df-dc 777
This theorem is referenced by: (None)
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