Theorem syl6ib

Description: A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 21-Jun-1993)

	Ref	Expression
Hypotheses	syl6ib.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
	syl6ib.2	⊢ ( 𝜒 ↔ 𝜃 )
Assertion	syl6ib	⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		syl6ib.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		syl6ib.2	⊢ ( 𝜒 ↔ 𝜃 )
3	2	biimpi	⊢ ( 𝜒 → 𝜃 )
4	1 3	syl6	⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )