Theorem ax3 1701

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 1692	. 2 ⊢ ((¬ 𝜑 → ¬ 𝜓) → (((¬ 𝜑 → 𝜑) → 𝜑) → (𝜓 → 𝜑)))
2		luklem4 1694	. 2 ⊢ ((((¬ 𝜑 → 𝜑) → 𝜑) → (𝜓 → 𝜑)) → (𝜓 → 𝜑))
3	1, 2	luklem1 1691	1 ⊢ ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))

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

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is used by: (None)