Theorem expt

Description: Exportation theorem pm3.3 (closed form of ex ) expressed with primitive connectives. (Contributed by NM, 28-Dec-1992) (Proof shortened by Garrett Katz, 25-May-2026)

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

Proof

Step	Hyp	Ref	Expression
1		pm3.2im	$$⊢\phi \to \left(\psi \to \neg \left(\phi \to \neg \psi \right)\right)$$
2		id	$$⊢\left(\neg \left(\phi \to \neg \psi \right)\to \chi \right)\to \left(\neg \left(\phi \to \neg \psi \right)\to \chi \right)$$
3	1 2	syl9r	$$⊢\left(\neg \left(\phi \to \neg \psi \right)\to \chi \right)\to \left(\phi \to \left(\psi \to \chi \right)\right)$$