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 → 𝜓 )
	tbsyl.2	⊢ ( 𝜓 → 𝜒 )
Assertion	tbsyl	⊢ ( 𝜑 → 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		tbsyl.1	⊢ ( 𝜑 → 𝜓 )
2		tbsyl.2	⊢ ( 𝜓 → 𝜒 )
3		tb-ax1	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) )
4	1 3	ax-mp	⊢ ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) )
5	2 4	ax-mp	⊢ ( 𝜑 → 𝜒 )