Theorem wl-luk-ax2

Description: ax-2 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-ax2	$$⊢\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\left(\phi \to \psi \right)\to \left(\phi \to \chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		wl-luk-pm2.21	$$⊢\neg \phi \to \left(\phi \to \chi \right)$$
2	1	wl-luk-a1d	$$⊢\neg \phi \to \left(\left(\phi \to \psi \right)\to \left(\phi \to \chi \right)\right)$$
3		wl-luk-imim2	$$⊢\left(\psi \to \chi \right)\to \left(\left(\phi \to \psi \right)\to \left(\phi \to \chi \right)\right)$$
4	2 3	wl-luk-ja	$$⊢\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\left(\phi \to \psi \right)\to \left(\phi \to \chi \right)\right)$$