Theorem wl-luk-ax3

Description: ax-3 proved from Lukasiewicz's axioms. (Contributed by Wolf Lammen, 17-Dec-2018) (New usage is discouraged.) (Proof modification is discouraged.)

	Ref	Expression
Assertion	wl-luk-ax3	$$⊢\left(\neg \phi \to \neg \psi \right)\to \left(\psi \to \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ax-luk3	$$⊢\psi \to \left(\neg \psi \to \phi \right)$$
2		ax-luk1	$$⊢\left(\neg \phi \to \neg \psi \right)\to \left(\left(\neg \psi \to \phi \right)\to \left(\neg \phi \to \phi \right)\right)$$
3	1 2	wl-luk-imtrid	$$⊢\left(\neg \phi \to \neg \psi \right)\to \left(\psi \to \left(\neg \phi \to \phi \right)\right)$$
4		ax-luk2	$$⊢\left(\neg \phi \to \phi \right)\to \phi$$
5	3 4	wl-luk-imtrdi	$$⊢\left(\neg \phi \to \neg \psi \right)\to \left(\psi \to \phi \right)$$