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Theorem bj-consensus 32891
Description: Version of consensus 1041 expressed using the conditional operator. (Remark: it may be better to express it as consensus 1041, 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 1058 . . 3 ((𝜓𝜒) → if-(𝜑, 𝜓, 𝜒))
21bj-jaoi2 32886 . 2 ((if-(𝜑, 𝜓, 𝜒) ∨ (𝜓𝜒)) → if-(𝜑, 𝜓, 𝜒))
3 orc 399 . 2 (if-(𝜑, 𝜓, 𝜒) → (if-(𝜑, 𝜓, 𝜒) ∨ (𝜓𝜒)))
42, 3impbii 199 1 ((if-(𝜑, 𝜓, 𝜒) ∨ (𝜓𝜒)) ↔ if-(𝜑, 𝜓, 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 382  wa 383  if-wif 1050
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 197  df-or 384  df-an 385  df-ifp 1051
This theorem is referenced by: (None)
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