 Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-dcbi GIF version

Theorem bj-dcbi 10048
 Description: Equivalence property for DECID. TODO: solve conflict with dcbi 844; minimize dcbii 747 and dcbid 748 with it, as well as theorems using those. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-dcbi ((𝜑𝜓) → (DECID 𝜑DECID 𝜓))

Proof of Theorem bj-dcbi
StepHypRef Expression
1 id 19 . . 3 ((𝜑𝜓) → (𝜑𝜓))
2 bj-notbi 10045 . . 3 ((𝜑𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
31, 2orbi12d 707 . 2 ((𝜑𝜓) → ((𝜑 ∨ ¬ 𝜑) ↔ (𝜓 ∨ ¬ 𝜓)))
4 df-dc 743 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
5 df-dc 743 . 2 (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
63, 4, 53bitr4g 212 1 ((𝜑𝜓) → (DECID 𝜑DECID 𝜓))
 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:  bj-d0clsepcl  10049
 Copyright terms: Public domain W3C validator