Theorem biadani

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

	Ref	Expression
Hypothesis	biadani.1	$$⊢\phi \to \psi$$
Assertion	biadani	$$⊢\left(\psi \to \left(\phi \leftrightarrow \chi \right)\right)\leftrightarrow \left(\phi \leftrightarrow \left(\psi \wedge \chi \right)\right)$$

Proof

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