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Theorem stabtestimpdc 823
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
StepHypRef Expression
1 exmiddc 743 . . . . . 6 (DECID ¬ φ → (¬ φ ¬ ¬ φ))
21adantl 262 . . . . 5 ((STAB φ DECID ¬ φ) → (¬ φ ¬ ¬ φ))
3 df-stab 739 . . . . . . . 8 (STAB φ ↔ (¬ ¬ φφ))
43biimpi 113 . . . . . . 7 (STAB φ → (¬ ¬ φφ))
54orim2d 701 . . . . . 6 (STAB φ → ((¬ φ ¬ ¬ φ) → (¬ φ φ)))
65adantr 261 . . . . 5 ((STAB φ DECID ¬ φ) → ((¬ φ ¬ ¬ φ) → (¬ φ φ)))
72, 6mpd 13 . . . 4 ((STAB φ DECID ¬ φ) → (¬ φ φ))
87orcomd 647 . . 3 ((STAB φ DECID ¬ φ) → (φ ¬ φ))
9 df-dc 742 . . 3 (DECID φ ↔ (φ ¬ φ))
108, 9sylibr 137 . 2 ((STAB φ DECID ¬ φ) → DECID φ)
11 dcimpstab 751 . . 3 (DECID φSTAB φ)
12 dcn 745 . . 3 (DECID φDECID ¬ φ)
1311, 12jca 290 . 2 (DECID φ → (STAB φ DECID ¬ φ))
1410, 13impbii 117 1 ((STAB φ DECID ¬ φ) ↔ DECID φ)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 628  STAB wstab 738  DECID wdc 741
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629
This theorem depends on definitions:  df-bi 110  df-stab 739  df-dc 742
This theorem is referenced by: (None)
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