Theorem wl-luk-ax3 38276

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 38264	. . 3 ⊢ (𝜓 → (¬ 𝜓 → 𝜑))
2		ax-luk1 38262	. . 3 ⊢ ((¬ 𝜑 → ¬ 𝜓) → ((¬ 𝜓 → 𝜑) → (¬ 𝜑 → 𝜑)))
3	1, 2	wl-luk-imtrid 38268	. 2 ⊢ ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → (¬ 𝜑 → 𝜑)))
4		ax-luk2 38263	. 2 ⊢ ((¬ 𝜑 → 𝜑) → 𝜑)
5	3, 4	wl-luk-imtrdi 38275	1 ⊢ ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4

This proof depends on axioms:  ax-mp 5  ax-luk1 38262  ax-luk2 38263  ax-luk3 38264

This theorem is used by:  wl-luk-ax1  38277