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	⊢ ( ( ¬ 𝜑 → ¬ 𝜓 ) → ( 𝜓 → 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		ax-luk3	⊢ ( 𝜓 → ( ¬ 𝜓 → 𝜑 ) )
2		ax-luk1	⊢ ( ( ¬ 𝜑 → ¬ 𝜓 ) → ( ( ¬ 𝜓 → 𝜑 ) → ( ¬ 𝜑 → 𝜑 ) ) )
3	1 2	wl-luk-imtrid	⊢ ( ( ¬ 𝜑 → ¬ 𝜓 ) → ( 𝜓 → ( ¬ 𝜑 → 𝜑 ) ) )
4		ax-luk2	⊢ ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 )
5	3 4	wl-luk-imtrdi	⊢ ( ( ¬ 𝜑 → ¬ 𝜓 ) → ( 𝜓 → 𝜑 ) )