Theorem syl8ib

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)

	Ref	Expression
Hypotheses	syl8ib.1	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) )
	syl8ib.2	⊢ ( 𝜃 ↔ 𝜏 )
Assertion	syl8ib	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜏 ) ) )

Proof

Step	Hyp	Ref	Expression
1		syl8ib.1	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) )
2		syl8ib.2	⊢ ( 𝜃 ↔ 𝜏 )
3	2	biimpi	⊢ ( 𝜃 → 𝜏 )
4	1 3	syl8	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜏 ) ) )