Theorem exbir

Description: Exportation implication also converting the consequent from a biconditional to an implication. Derived automatically from exbirVD . (Contributed by Alan Sare, 31-Dec-2011)

	Ref	Expression
Assertion	exbir	$$⊢\left(\left(\phi \wedge \psi \right)\to \left(\chi \leftrightarrow \theta \right)\right)\to \left(\phi \to \left(\psi \to \left(\theta \to \chi \right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		biimpr	$$⊢\left(\chi \leftrightarrow \theta \right)\to \left(\theta \to \chi \right)$$
2	1	imim2i	$$⊢\left(\left(\phi \wedge \psi \right)\to \left(\chi \leftrightarrow \theta \right)\right)\to \left(\left(\phi \wedge \psi \right)\to \left(\theta \to \chi \right)\right)$$
3	2	expd	$$⊢\left(\left(\phi \wedge \psi \right)\to \left(\chi \leftrightarrow \theta \right)\right)\to \left(\phi \to \left(\psi \to \left(\theta \to \chi \right)\right)\right)$$