Theorem bj-consensusALT 37057

Description: Alternate proof of bj-consensus 37056. (Contributed by BJ, 30-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-consensusALT	⊢ ((if-(𝜑, 𝜓, 𝜒) ∨ (𝜓 ∧ 𝜒)) ↔ if-(𝜑, 𝜓, 𝜒))

Proof of Theorem bj-consensusALT

Step	Hyp	Ref	Expression
1		orcom 883	. 2 ⊢ ((if-(𝜑, 𝜓, 𝜒) ∨ (𝜓 ∧ 𝜒)) ↔ ((𝜓 ∧ 𝜒) ∨ if-(𝜑, 𝜓, 𝜒)))
2		anifp 1086	. . 3 ⊢ ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒))
3		pm4.72 964	. . 3 ⊢ (((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒)) ↔ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜓 ∧ 𝜒) ∨ if-(𝜑, 𝜓, 𝜒))))
4	2, 3	mpbi 233	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜓 ∧ 𝜒) ∨ if-(𝜑, 𝜓, 𝜒)))
5	1, 4	bitr4i 281	1 ⊢ ((if-(𝜑, 𝜓, 𝜒) ∨ (𝜓 ∧ 𝜒)) ↔ if-(𝜑, 𝜓, 𝜒))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860  if-wif 1076

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 210  df-an 401  df-or 861  df-ifp 1077

This theorem is referenced by: (None)