Theorem bj-consensus 34686

Description: Version of consensus 1049 expressed using the conditional operator. (Remark: it may be better to express it as consensus 1049, 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) ∨ (𝜓 ∧ 𝜒)) ↔ if-(𝜑, 𝜓, 𝜒))

Proof of Theorem bj-consensus

Step	Hyp	Ref	Expression
1		anifp 1068	. . 3 ⊢ ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒))
2	1	bj-jaoi2 34680	. 2 ⊢ ((if-(𝜑, 𝜓, 𝜒) ∨ (𝜓 ∧ 𝜒)) → if-(𝜑, 𝜓, 𝜒))
3		orc 863	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) → (if-(𝜑, 𝜓, 𝜒) ∨ (𝜓 ∧ 𝜒)))
4	2, 3	impbii 208	1 ⊢ ((if-(𝜑, 𝜓, 𝜒) ∨ (𝜓 ∧ 𝜒)) ↔ if-(𝜑, 𝜓, 𝜒))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∨ wo 843  if-wif 1059

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 206  df-an 396  df-or 844  df-ifp 1060

This theorem is referenced by: (None)