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	$$⊢\phi \to \psi$$
	tbsyl.2	$$⊢\psi \to \chi$$
Assertion	tbsyl	$$⊢\phi \to \chi$$

Proof

Step	Hyp	Ref	Expression
1		tbsyl.1	$$⊢\phi \to \psi$$
2		tbsyl.2	$$⊢\psi \to \chi$$
3		tb-ax1	$$⊢\left(\phi \to \psi \right)\to \left(\left(\psi \to \chi \right)\to \left(\phi \to \chi \right)\right)$$
4	1 3	ax-mp	$$⊢\left(\psi \to \chi \right)\to \left(\phi \to \chi \right)$$
5	2 4	ax-mp	$$⊢\phi \to \chi$$