Theorem baibd

Description: Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015)

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

Proof

Step	Hyp	Ref	Expression
1		baibd.1	$$⊢\phi \to \left(\psi \leftrightarrow \left(\chi \wedge \theta \right)\right)$$
2		ibar	$$⊢\chi \to \left(\theta \leftrightarrow \left(\chi \wedge \theta \right)\right)$$
3	2	bicomd	$$⊢\chi \to \left(\left(\chi \wedge \theta \right)\leftrightarrow \theta \right)$$
4	1 3	sylan9bb	$$⊢\left(\phi \wedge \chi \right)\to \left(\psi \leftrightarrow \theta \right)$$