Theorem mpbirand

Description: Detach truth from conjunction in biconditional. (Contributed by Glauco Siliprandi, 3-Mar-2021)

	Ref	Expression
Hypotheses	mpbirand.1	⊢ ( 𝜑 → 𝜒 )
	mpbirand.2	⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜒 ∧ 𝜃 ) ) )
Assertion	mpbirand	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		mpbirand.1	⊢ ( 𝜑 → 𝜒 )
2		mpbirand.2	⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜒 ∧ 𝜃 ) ) )
3	1	biantrurd	⊢ ( 𝜑 → ( 𝜃 ↔ ( 𝜒 ∧ 𝜃 ) ) )
4	2 3	bitr4d	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜃 ) )