Theorem syl8

Description: A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 1-Aug-1994) (Proof shortened by Wolf Lammen, 3-Aug-2012)

	Ref	Expression
Hypotheses	syl8.1	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) )
	syl8.2	⊢ ( 𝜃 → 𝜏 )
Assertion	syl8	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜏 ) ) )

Proof

Step	Hyp	Ref	Expression
1		syl8.1	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) )
2		syl8.2	⊢ ( 𝜃 → 𝜏 )
3	2	a1i	⊢ ( 𝜑 → ( 𝜃 → 𝜏 ) )
4	1 3	syl6d	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜏 ) ) )