Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-consensus Structured version   Visualization version   GIF version

Theorem bj-consensus 33808
Description: Version of consensus 1044 expressed using the conditional operator. (Remark: it may be better to express it as consensus 1044, using only binary connectives, and hinting at the fact that it is a Boolean algebra identity, like the absorption identities.) (Contributed by BJ, 30-Sep-2019.)
Assertion
Ref Expression
bj-consensus ((if-(𝜑, 𝜓, 𝜒) ∨ (𝜓𝜒)) ↔ if-(𝜑, 𝜓, 𝜒))

Proof of Theorem bj-consensus
StepHypRef Expression
1 anifp 1062 . . 3 ((𝜓𝜒) → if-(𝜑, 𝜓, 𝜒))
21bj-jaoi2 33802 . 2 ((if-(𝜑, 𝜓, 𝜒) ∨ (𝜓𝜒)) → if-(𝜑, 𝜓, 𝜒))
3 orc 861 . 2 (if-(𝜑, 𝜓, 𝜒) → (if-(𝜑, 𝜓, 𝜒) ∨ (𝜓𝜒)))
42, 3impbii 210 1 ((if-(𝜑, 𝜓, 𝜒) ∨ (𝜓𝜒)) ↔ if-(𝜑, 𝜓, 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wo 841  if-wif 1054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ifp 1055
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator