Theorem imdistan

Description: Distribution of implication with conjunction. (Contributed by NM, 31-May-1999) (Proof shortened by Wolf Lammen, 6-Dec-2012)

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

Proof

Step	Hyp	Ref	Expression
1		pm5.42	$$⊢\left(\phi \to \left(\psi \to \chi \right)\right)\leftrightarrow \left(\phi \to \left(\psi \to \left(\phi \wedge \chi \right)\right)\right)$$
2		impexp	$$⊢\left(\left(\phi \wedge \psi \right)\to \left(\phi \wedge \chi \right)\right)\leftrightarrow \left(\phi \to \left(\psi \to \left(\phi \wedge \chi \right)\right)\right)$$
3	1 2	bitr4i	$$⊢\left(\phi \to \left(\psi \to \chi \right)\right)\leftrightarrow \left(\left(\phi \wedge \psi \right)\to \left(\phi \wedge \chi \right)\right)$$