Theorem biadani

Description: Inference associated with biadan . (Contributed by BJ, 4-Mar-2023)

	Ref	Expression
Hypothesis	biadani.1	⊢ ( 𝜑 → 𝜓 )
Assertion	biadani	⊢ ( ( 𝜓 → ( 𝜑 ↔ 𝜒 ) ) ↔ ( 𝜑 ↔ ( 𝜓 ∧ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		biadani.1	⊢ ( 𝜑 → 𝜓 )
2		biadan	⊢ ( ( 𝜑 → 𝜓 ) ↔ ( ( 𝜓 → ( 𝜑 ↔ 𝜒 ) ) ↔ ( 𝜑 ↔ ( 𝜓 ∧ 𝜒 ) ) ) )
3	1 2	mpbi	⊢ ( ( 𝜓 → ( 𝜑 ↔ 𝜒 ) ) ↔ ( 𝜑 ↔ ( 𝜓 ∧ 𝜒 ) ) )