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Theorem dcbi 855
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 795 . . 3 (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
2 dcim 795 . . . 4 (DECID 𝜓 → (DECID 𝜑DECID (𝜓𝜑)))
32com12 30 . . 3 (DECID 𝜑 → (DECID 𝜓DECID (𝜓𝜑)))
4 dcan 853 . . 3 (DECID (𝜑𝜓) → (DECID (𝜓𝜑) → DECID ((𝜑𝜓) ∧ (𝜓𝜑))))
51, 3, 4syl6c 64 . 2 (DECID 𝜑 → (DECID 𝜓DECID ((𝜑𝜓) ∧ (𝜓𝜑))))
6 dfbi2 374 . . 3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜑)))
76dcbii 758 . 2 (DECID (𝜑𝜓) ↔ DECID ((𝜑𝜓) ∧ (𝜓𝜑)))
85, 7syl6ibr 155 1 (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  DECID wdc 753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-dc 754
This theorem is referenced by:  xor3dc  1294  pm5.15dc  1296  bilukdc  1303  xordidc  1306
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