Theorem ax3 1671

Description: Standard propositional axiom derived from Lukasiewicz axioms. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax3	⊢ ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))

Proof of Theorem ax3

Step	Hyp	Ref	Expression
1		luklem2 1662	. 2 ⊢ ((¬ 𝜑 → ¬ 𝜓) → (((¬ 𝜑 → 𝜑) → 𝜑) → (𝜓 → 𝜑)))
2		luklem4 1664	. 2 ⊢ ((((¬ 𝜑 → 𝜑) → 𝜑) → (𝜓 → 𝜑)) → (𝜓 → 𝜑))
3	1, 2	luklem1 1661	1 ⊢ ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is referenced by: (None)