Theorem bj-consensus 37228

Description: Version of consensus 1068 expressed using the conditional operator. (Remark: it may be better to express it as consensus 1068, 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 1088	. . 3 ⊢ ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒))
2	1	bj-jaoi2 37222	. 2 ⊢ ((if-(𝜑, 𝜓, 𝜒) ∨ (𝜓 ∧ 𝜒)) → if-(𝜑, 𝜓, 𝜒))
3		orc 881	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) → (if-(𝜑, 𝜓, 𝜒) ∨ (𝜓 ∧ 𝜒)))
4	2, 3	impbii 212	1 ⊢ ((if-(𝜑, 𝜓, 𝜒) ∨ (𝜓 ∧ 𝜒)) ↔ if-(𝜑, 𝜓, 𝜒))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∨ wo 861  if-wif 1078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079

This theorem is used by: (None)