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Metamath Proof Explorer

Theorem imbitrrdi

Description: A mixed syllogism inference from a nested implication and a biconditional. Useful for substituting an embedded consequent with a definition. (Contributed by NM, 5-Aug-1993)

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

Proof

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