Theorem sylbi

Description: A mixed syllogism inference from a biconditional and an implication. Useful for substituting an antecedent with a definition. (Contributed by NM, 3-Jan-1993)

	Ref	Expression
Hypotheses	sylbi.1	$$⊢\phi \leftrightarrow \psi$$
	sylbi.2	$$⊢\psi \to \chi$$
Assertion	sylbi	$$⊢\phi \to \chi$$

Proof

Step	Hyp	Ref	Expression
1		sylbi.1	$$⊢\phi \leftrightarrow \psi$$
2		sylbi.2	$$⊢\psi \to \chi$$
3	1	biimpi	$$⊢\phi \to \psi$$
4	3 2	syl	$$⊢\phi \to \chi$$