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Theorem dcbi 844
 Description: An equivalence of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.)
Assertion
Ref Expression
dcbi (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))

Proof of Theorem dcbi
StepHypRef Expression
1 dcim 784 . . 3 (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
2 dcim 784 . . . 4 (DECID 𝜓 → (DECID 𝜑DECID (𝜓𝜑)))
32com12 27 . . 3 (DECID 𝜑 → (DECID 𝜓DECID (𝜓𝜑)))
4 dcan 842 . . 3 (DECID (𝜑𝜓) → (DECID (𝜓𝜑) → DECID ((𝜑𝜓) ∧ (𝜓𝜑))))
51, 3, 4syl6c 60 . 2 (DECID 𝜑 → (DECID 𝜓DECID ((𝜑𝜓) ∧ (𝜓𝜑))))
6 dfbi2 368 . . 3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜑)))
76dcbii 747 . 2 (DECID (𝜑𝜓) ↔ DECID ((𝜑𝜓) ∧ (𝜓𝜑)))
85, 7syl6ibr 151 1 (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  DECID wdc 742 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 630 This theorem depends on definitions:  df-bi 110  df-dc 743 This theorem is referenced by:  xor3dc  1278  pm5.15dc  1280  bilukdc  1287  xordidc  1290
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