Theorem imimorb

Description: Simplify an implication between implications. (Contributed by Paul Chapman, 17-Nov-2012) (Proof shortened by Wolf Lammen, 3-Apr-2013)

	Ref	Expression
Assertion	imimorb	$$⊢\left(\left(\psi \to \chi \right)\to \left(\phi \to \chi \right)\right)\leftrightarrow \left(\phi \to \left(\psi \vee \chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		bi2.04	$$⊢\left(\left(\psi \to \chi \right)\to \left(\phi \to \chi \right)\right)\leftrightarrow \left(\phi \to \left(\left(\psi \to \chi \right)\to \chi \right)\right)$$
2		dfor2	$$⊢\left(\psi \vee \chi \right)\leftrightarrow \left(\left(\psi \to \chi \right)\to \chi \right)$$
3	2	imbi2i	$$⊢\left(\phi \to \left(\psi \vee \chi \right)\right)\leftrightarrow \left(\phi \to \left(\left(\psi \to \chi \right)\to \chi \right)\right)$$
4	1 3	bitr4i	$$⊢\left(\left(\psi \to \chi \right)\to \left(\phi \to \chi \right)\right)\leftrightarrow \left(\phi \to \left(\psi \vee \chi \right)\right)$$