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Theorem dcbi 880
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 820 . . 3 (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
2 dcim 820 . . . 4 (DECID 𝜓 → (DECID 𝜑DECID (𝜓𝜑)))
32com12 30 . . 3 (DECID 𝜑 → (DECID 𝜓DECID (𝜓𝜑)))
4 dcan 878 . . 3 (DECID (𝜑𝜓) → (DECID (𝜓𝜑) → DECID ((𝜑𝜓) ∧ (𝜓𝜑))))
51, 3, 4syl6c 65 . 2 (DECID 𝜑 → (DECID 𝜓DECID ((𝜑𝜓) ∧ (𝜓𝜑))))
6 dfbi2 380 . . 3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜑)))
76dcbii 783 . 2 (DECID (𝜑𝜓) ↔ DECID ((𝜑𝜓) ∧ (𝜓𝜑)))
85, 7syl6ibr 160 1 (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  DECID wdc 778
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 779
This theorem is referenced by:  xor3dc  1321  pm5.15dc  1323  bilukdc  1330  xordidc  1333
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