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

Theorem bj-dcbi 10414
Description: Equivalence property for DECID. TODO: solve conflict with dcbi 855; minimize dcbii 758 and dcbid 759 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 10411 . . 3 ((𝜑𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
31, 2orbi12d 717 . 2 ((𝜑𝜓) → ((𝜑 ∨ ¬ 𝜑) ↔ (𝜓 ∨ ¬ 𝜓)))
4 df-dc 754 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
5 df-dc 754 . 2 (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
63, 4, 53bitr4g 216 1 ((𝜑𝜓) → (DECID 𝜑DECID 𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 102  wo 639  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:  bj-d0clsepcl  10415
  Copyright terms: Public domain W3C validator