Theorem impt

Description: Importation theorem pm3.1 (closed form of imp ) expressed with primitive connectives. (Contributed by NM, 25-Apr-1994) (Proof shortened by Wolf Lammen, 20-Jul-2013)

	Ref	Expression
Assertion	impt	$$⊢\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\neg \left(\phi \to \neg \psi \right)\to \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		simprim	$$⊢\neg \left(\phi \to \neg \psi \right)\to \psi$$
2		simplim	$$⊢\neg \left(\phi \to \neg \psi \right)\to \phi$$
3	2	imim1i	$$⊢\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\neg \left(\phi \to \neg \psi \right)\to \left(\psi \to \chi \right)\right)$$
4	1 3	mpdi	$$⊢\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\neg \left(\phi \to \neg \psi \right)\to \chi \right)$$