Theorem syl6mpi

Description: A syllogism inference. (Contributed by Alan Sare, 8-Jul-2011) (Proof shortened by Wolf Lammen, 13-Sep-2012)

	Ref	Expression
Hypotheses	syl6mpi.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
	syl6mpi.2	⊢ 𝜃
	syl6mpi.3	⊢ ( 𝜒 → ( 𝜃 → 𝜏 ) )
Assertion	syl6mpi	⊢ ( 𝜑 → ( 𝜓 → 𝜏 ) )

Proof

Step	Hyp	Ref	Expression
1		syl6mpi.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		syl6mpi.2	⊢ 𝜃
3		syl6mpi.3	⊢ ( 𝜒 → ( 𝜃 → 𝜏 ) )
4	2 3	mpi	⊢ ( 𝜒 → 𝜏 )
5	1 4	syl6	⊢ ( 𝜑 → ( 𝜓 → 𝜏 ) )