Theorem sylnbi

Description: A mixed syllogism inference from a biconditional and an implication. Useful for substituting an antecedent with a definition. (Contributed by Wolf Lammen, 16-Dec-2013)

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

Proof

Step	Hyp	Ref	Expression
1		sylnbi.1	$$⊢\phi \leftrightarrow \psi$$
2		sylnbi.2	$$⊢\neg \psi \to \chi$$
3	1	notbii	$$⊢\neg \phi \leftrightarrow \neg \psi$$
4	3 2	sylbi	$$⊢\neg \phi \to \chi$$