Theorem wl-ax3 32431

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-ax3	⊢ ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))

Proof of Theorem wl-ax3

Step	Hyp	Ref	Expression
1		ax-luk3 32419	. . 3 ⊢ (𝜓 → (¬ 𝜓 → 𝜑))
2		ax-luk1 32417	. . 3 ⊢ ((¬ 𝜑 → ¬ 𝜓) → ((¬ 𝜓 → 𝜑) → (¬ 𝜑 → 𝜑)))
3	1, 2	wl-syl5 32423	. 2 ⊢ ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → (¬ 𝜑 → 𝜑)))
4		ax-luk2 32418	. 2 ⊢ ((¬ 𝜑 → 𝜑) → 𝜑)
5	3, 4	wl-syl6 32430	1 ⊢ ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-luk1 32417  ax-luk2 32418  ax-luk3 32419

This theorem is referenced by:  wl-ax1  32432