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Theorem stdcndc 813
Description: A formula is decidable if and only if its negation is decidable and it is stable (that is, it is testable and stable). (Contributed by David A. Wheeler, 13-Aug-2018.) (Proof shortened by BJ, 28-Oct-2023.)
Assertion
Ref Expression
stdcndc ((STAB 𝜑DECID ¬ 𝜑) ↔ DECID 𝜑)

Proof of Theorem stdcndc
StepHypRef Expression
1 df-stab 799 . . . 4 (STAB 𝜑 ↔ (¬ ¬ 𝜑𝜑))
2 df-dc 803 . . . 4 (DECID ¬ 𝜑 ↔ (¬ 𝜑 ∨ ¬ ¬ 𝜑))
3 pm2.36 776 . . . . 5 ((¬ ¬ 𝜑𝜑) → ((¬ 𝜑 ∨ ¬ ¬ 𝜑) → (𝜑 ∨ ¬ 𝜑)))
43imp 123 . . . 4 (((¬ ¬ 𝜑𝜑) ∧ (¬ 𝜑 ∨ ¬ ¬ 𝜑)) → (𝜑 ∨ ¬ 𝜑))
51, 2, 4syl2anb 287 . . 3 ((STAB 𝜑DECID ¬ 𝜑) → (𝜑 ∨ ¬ 𝜑))
6 df-dc 803 . . 3 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
75, 6sylibr 133 . 2 ((STAB 𝜑DECID ¬ 𝜑) → DECID 𝜑)
8 dcstab 812 . . 3 (DECID 𝜑STAB 𝜑)
9 dcn 810 . . 3 (DECID 𝜑DECID ¬ 𝜑)
108, 9jca 302 . 2 (DECID 𝜑 → (STAB 𝜑DECID ¬ 𝜑))
117, 10impbii 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:  stdcn  815
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