Theorem wl-luk-ax3 35531

Description: ax-3 8 proved from Lukasiewicz's axioms. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
wl-luk-ax3	⊢ ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))

Proof of Theorem wl-luk-ax3

Step	Hyp	Ref	Expression
1		ax-luk3 35519	. . 3 ⊢ (𝜓 → (¬ 𝜓 → 𝜑))
2		ax-luk1 35517	. . 3 ⊢ ((¬ 𝜑 → ¬ 𝜓) → ((¬ 𝜓 → 𝜑) → (¬ 𝜑 → 𝜑)))
3	1, 2	wl-luk-imtrid 35523	. 2 ⊢ ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → (¬ 𝜑 → 𝜑)))
4		ax-luk2 35518	. 2 ⊢ ((¬ 𝜑 → 𝜑) → 𝜑)
5	3, 4	wl-luk-imtrdi 35530	1 ⊢ ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-luk1 35517  ax-luk2 35518  ax-luk3 35519

This theorem is referenced by:  wl-luk-ax1  35532