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

Theorem dcbid 748
 Description: The equivalent of a decidable proposition is decidable. (Contributed by Jim Kingdon, 7-Sep-2019.)
Hypothesis
Ref Expression
dcbid.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
dcbid (𝜑 → (DECID 𝜓DECID 𝜒))

Proof of Theorem dcbid
StepHypRef Expression
1 dcbid.1 . . 3 (𝜑 → (𝜓𝜒))
21notbid 592 . . 3 (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))
31, 2orbi12d 707 . 2 (𝜑 → ((𝜓 ∨ ¬ 𝜓) ↔ (𝜒 ∨ ¬ 𝜒)))
4 df-dc 743 . 2 (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
5 df-dc 743 . 2 (DECID 𝜒 ↔ (𝜒 ∨ ¬ 𝜒))
63, 4, 53bitr4g 212 1 (𝜑 → (DECID 𝜓DECID 𝜒))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98   ∨ wo 629  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:  ltdcpi  6421  enqdc  6459  enqdc1  6460  ltdcnq  6495
 Copyright terms: Public domain W3C validator