Theorem luklem1 1690

Description: Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 23-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
luklem1.1	⊢ (𝜑 → 𝜓)
luklem1.2	⊢ (𝜓 → 𝜒)

Assertion

Ref	Expression
luklem1	⊢ (𝜑 → 𝜒)

Proof of Theorem luklem1

Step	Hyp	Ref	Expression
1		luklem1.2	. 2 ⊢ (𝜓 → 𝜒)
2		luklem1.1	. . 3 ⊢ (𝜑 → 𝜓)
3		luk-1 1687	. . 3 ⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))
4	2, 3	ax-mp 5	. 2 ⊢ ((𝜓 → 𝜒) → (𝜑 → 𝜒))
5	1, 4	ax-mp 5	1 ⊢ (𝜑 → 𝜒)

This proof depends on syntax axioms:   → wi 4

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

This theorem is used by:  luklem2  1691  luklem3  1692  luklem4  1693  luklem5  1694  luklem6  1695  luklem7  1696  ax2  1699  ax3  1700