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	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		wl-luk-pm2.21	⊢ ( ¬ 𝜑 → ( 𝜑 → 𝜒 ) )
2	1	wl-luk-a1d	⊢ ( ¬ 𝜑 → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) )
3		wl-luk-imim2	⊢ ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) )
4	2 3	wl-luk-ja	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) )