This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem tbsyl

Description: The weak syllogism from Tarski-Bernays'. (Contributed by Anthony Hart, 16-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		tbsyl.1	$⊢ φ \to ψ$
2		tbsyl.2	$⊢ ψ \to χ$
3		tb-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	$⊢ φ \to χ$