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Theorem bj-consensus 33141
 Description: Version of consensus 1036 expressed using the conditional operator. (Remark: it may be better to express it as consensus 1036, 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 1054 . . 3 ((𝜓𝜒) → if-(𝜑, 𝜓, 𝜒))
21bj-jaoi2 33135 . 2 ((if-(𝜑, 𝜓, 𝜒) ∨ (𝜓𝜒)) → if-(𝜑, 𝜓, 𝜒))
3 orc 856 . 2 (if-(𝜑, 𝜓, 𝜒) → (if-(𝜑, 𝜓, 𝜒) ∨ (𝜓𝜒)))
42, 3impbii 201 1 ((if-(𝜑, 𝜓, 𝜒) ∨ (𝜓𝜒)) ↔ if-(𝜑, 𝜓, 𝜒))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 386   ∨ wo 836  if-wif 1046 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 199  df-an 387  df-or 837  df-ifp 1047 This theorem is referenced by: (None)
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