Theorem sylnibr

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

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

Proof

Step	Hyp	Ref	Expression
1		sylnibr.1	$$⊢\phi \to \neg \psi$$
2		sylnibr.2	$$⊢\chi \leftrightarrow \psi$$
3	2	bicomi	$$⊢\psi \leftrightarrow \chi$$
4	1 3	sylnib	$$⊢\phi \to \neg \chi$$