Theorem syl6bir

Description: A mixed syllogism inference. (Contributed by NM, 18-May-1994)

	Ref	Expression
Hypotheses	syl6bir.1	⊢ ( 𝜑 → ( 𝜒 ↔ 𝜓 ) )
	syl6bir.2	⊢ ( 𝜒 → 𝜃 )
Assertion	syl6bir	⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		syl6bir.1	⊢ ( 𝜑 → ( 𝜒 ↔ 𝜓 ) )
2		syl6bir.2	⊢ ( 𝜒 → 𝜃 )
3	1	biimprd	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
4	3 2	syl6	⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )