Theorem wl-luk-ax1

Description: ax-1 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-ax1	⊢ ( 𝜑 → ( 𝜓 → 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		ax-luk3	⊢ ( 𝜑 → ( ¬ 𝜑 → ¬ 𝜓 ) )
2		wl-luk-ax3	⊢ ( ( ¬ 𝜑 → ¬ 𝜓 ) → ( 𝜓 → 𝜑 ) )
3	1 2	wl-luk-syl	⊢ ( 𝜑 → ( 𝜓 → 𝜑 ) )