Theorem sylibr

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

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

Proof

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