Theorem wl-luk-ax1 37419

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 37406	. 2 ⊢ (𝜑 → (¬ 𝜑 → ¬ 𝜓))
2		wl-luk-ax3 37418	. 2 ⊢ ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))
3	1, 2	wl-luk-syl 37409	1 ⊢ (𝜑 → (𝜓 → 𝜑))

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-luk1 37404  ax-luk2 37405  ax-luk3 37406

This theorem is referenced by:  wl-luk-pm2.27  37420  wl-luk-a1d  37426  wl-luk-pm2.04  37430