ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dcbi GIF version

Theorem dcbi 903
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 809 . . 3 (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
2 dcim 809 . . . 4 (DECID 𝜓 → (DECID 𝜑DECID (𝜓𝜑)))
32com12 30 . . 3 (DECID 𝜑 → (DECID 𝜓DECID (𝜓𝜑)))
4 dcan 901 . . 3 (DECID (𝜑𝜓) → (DECID (𝜓𝜑) → DECID ((𝜑𝜓) ∧ (𝜓𝜑))))
51, 3, 4syl6c 66 . 2 (DECID 𝜑 → (DECID 𝜓DECID ((𝜑𝜓) ∧ (𝜓𝜑))))
6 dfbi2 383 . . 3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜑)))
76dcbii 808 . 2 (DECID (𝜑𝜓) ↔ DECID ((𝜑𝜓) ∧ (𝜓𝜑)))
85, 7syl6ibr 161 1 (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  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-dc 803
This theorem is referenced by:  xor3dc  1348  pm5.15dc  1350  bilukdc  1357  xordidc  1360
  Copyright terms: Public domain W3C validator