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Metamath Proof Explorer

Theorem tbwsyl

Description: Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	tbwsyl.1	$⊢ φ \to ψ$
	tbwsyl.2	$⊢ ψ \to χ$
Assertion	tbwsyl	$⊢ φ \to χ$

Proof

Step	Hyp	Ref	Expression
1		tbwsyl.1	$⊢ φ \to ψ$
2		tbwsyl.2	$⊢ ψ \to χ$
3		tbw-ax1	$⊢ (φ \to ψ) \to ((ψ \to χ) \to (φ \to χ))$
4	1 3	ax-mp	$⊢ (ψ \to χ) \to (φ \to χ)$
5	2 4	ax-mp	$⊢ φ \to χ$