Theorem wl-luk-ax2 38116

Description: ax-2 7 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-ax2	⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))

Proof of Theorem wl-luk-ax2

Step	Hyp	Ref	Expression
1		wl-luk-pm2.21 38111	. . 3 ⊢ (¬ 𝜑 → (𝜑 → 𝜒))
2	1	wl-luk-a1d 38115	. 2 ⊢ (¬ 𝜑 → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
3		wl-luk-imim2 38114	. 2 ⊢ ((𝜓 → 𝜒) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
4	2, 3	wl-luk-ja 38113	1 ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))

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-pm2.04  38119