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	$$⊢\phi \to \psi$$
	biadanii.2	$$⊢\psi \to \left(\phi \leftrightarrow \chi \right)$$
Assertion	biadanii	$$⊢\phi \leftrightarrow \left(\psi \wedge \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		biadani.1	$$⊢\phi \to \psi$$
2		biadanii.2	$$⊢\psi \to \left(\phi \leftrightarrow \chi \right)$$
3	1	biadani	$$⊢\left(\psi \to \left(\phi \leftrightarrow \chi \right)\right)\leftrightarrow \left(\phi \leftrightarrow \left(\psi \wedge \chi \right)\right)$$
4	2 3	mpbi	$$⊢\phi \leftrightarrow \left(\psi \wedge \chi \right)$$