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Metamath Proof Explorer

Theorem biimtrid

Description: A mixed syllogism inference from a nested implication and a biconditional. Useful for substituting an embedded antecedent with a definition. (Contributed by NM, 12-Jan-1993)

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

Proof

Step	Hyp	Ref	Expression
1		biimtrid.1	⊢ ( 𝜑 ↔ 𝜓 )
2		biimtrid.2	⊢ ( 𝜒 → ( 𝜓 → 𝜃 ) )
3	1	biimpi	⊢ ( 𝜑 → 𝜓 )
4	3 2	syl5	⊢ ( 𝜒 → ( 𝜑 → 𝜃 ) )