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Theorem stabtestimpdc 745
Description: "Stable and testable" is equivalent to decidable. (Contributed by David A. Wheeler, 13-Aug-2018.)
Assertion
Ref Expression
stabtestimpdc ((STAB φ TEST φ) ↔ DECID φ)

Proof of Theorem stabtestimpdc
StepHypRef Expression
1 df-test 731 . . . . . . 7 (TEST φ ↔ (¬ φ ¬ ¬ φ))
21biimpi 113 . . . . . 6 (TEST φ → (¬ φ ¬ ¬ φ))
32adantl 262 . . . . 5 ((STAB φ TEST φ) → (¬ φ ¬ ¬ φ))
4 df-stab 728 . . . . . . . 8 (STAB φ ↔ (¬ ¬ φφ))
54biimpi 113 . . . . . . 7 (STAB φ → (¬ ¬ φφ))
65orim2d 689 . . . . . 6 (STAB φ → ((¬ φ ¬ ¬ φ) → (¬ φ φ)))
76adantr 261 . . . . 5 ((STAB φ TEST φ) → ((¬ φ ¬ ¬ φ) → (¬ φ φ)))
83, 7mpd 13 . . . 4 ((STAB φ TEST φ) → (¬ φ φ))
98orcomd 635 . . 3 ((STAB φ TEST φ) → (φ ¬ φ))
10 df-dc 734 . . 3 (DECID φ ↔ (φ ¬ φ))
119, 10sylibr 137 . 2 ((STAB φ TEST φ) → DECID φ)
12 dcimpstab 743 . . 3 (DECID φSTAB φ)
13 dcimptest 744 . . 3 (DECID φTEST φ)
1412, 13jca 290 . 2 (DECID φ → (STAB φ TEST φ))
1511, 14impbii 117 1 ((STAB φ TEST φ) ↔ DECID φ)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 616  STAB wstab 727  TEST wtest 730  DECID wdc 733
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 532  ax-in2 533  ax-io 617
This theorem depends on definitions:  df-bi 110  df-stab 728  df-test 731  df-dc 734
This theorem is referenced by: (None)
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