Theorem biadanii

Description: Inference associated with biadani . Add a conjunction to an equivalence. (Contributed by Jeff Madsen, 20-Jun-2011) (Proof shortened by BJ, 4-Mar-2023)

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

Proof

Step	Hyp	Ref	Expression
1		biadani.1	⊢ ( 𝜑 → 𝜓 )
2		biadanii.2	⊢ ( 𝜓 → ( 𝜑 ↔ 𝜒 ) )
3	1	biadani	⊢ ( ( 𝜓 → ( 𝜑 ↔ 𝜒 ) ) ↔ ( 𝜑 ↔ ( 𝜓 ∧ 𝜒 ) ) )
4	2 3	mpbi	⊢ ( 𝜑 ↔ ( 𝜓 ∧ 𝜒 ) )