Theorem wl-luk-ax1 38108

Description: ax-1 6 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-ax1	⊢ (𝜑 → (𝜓 → 𝜑))

Proof of Theorem wl-luk-ax1

Step	Hyp	Ref	Expression
1		ax-luk3 38095	. 2 ⊢ (𝜑 → (¬ 𝜑 → ¬ 𝜓))
2		wl-luk-ax3 38107	. 2 ⊢ ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))
3	1, 2	wl-luk-syl 38098	1 ⊢ (𝜑 → (𝜓 → 𝜑))

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

This proof depends on axioms:  ax-mp 5  ax-luk1 38093  ax-luk2 38094  ax-luk3 38095

This theorem is used by:  wl-luk-pm2.27  38109  wl-luk-a1d  38115  wl-luk-pm2.04  38119