Theorem syl6ibr

Description: A mixed syllogism inference from a nested implication and a biconditional. Useful for substituting an embedded consequent with a definition. In the process of being renamed to imbitrrdi . (Contributed by NM, 10-Jan-1993)

	Ref	Expression
Hypotheses	syl6ibr.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
	syl6ibr.2	⊢ ( 𝜃 ↔ 𝜒 )
Assertion	syl6ibr	⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		syl6ibr.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		syl6ibr.2	⊢ ( 𝜃 ↔ 𝜒 )
3	2	biimpri	⊢ ( 𝜒 → 𝜃 )
4	1 3	syl6	⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )