Theorem luklem1

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

	Ref	Expression
Hypotheses	luklem1.1	$$⊢\phi \to \psi$$
	luklem1.2	$$⊢\psi \to \chi$$
Assertion	luklem1	$$⊢\phi \to \chi$$

Proof

Step	Hyp	Ref	Expression
1		luklem1.1	$$⊢\phi \to \psi$$
2		luklem1.2	$$⊢\psi \to \chi$$
3		luk-1	$$⊢\left(\phi \to \psi \right)\to \left(\left(\psi \to \chi \right)\to \left(\phi \to \chi \right)\right)$$
4	1 3	ax-mp	$$⊢\left(\psi \to \chi \right)\to \left(\phi \to \chi \right)$$
5	2 4	ax-mp	$$⊢\phi \to \chi$$