Theorem bj-consensusALT

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

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

Proof

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