The most reliable method for how the validity of a logical syllogism can be verified is to formalize it and show that there is either a formal proof or it is true in any model. A specific method for proving validity is to use special rules that have been used by logicians. However, we cannot be sure that they indeed verify the validity of syllogisms. The goal of this presentation is to show that the rules indeed work.  Peterson studied syllogisms with intermediate quantifiers and suggested extension of the rules also to them. In this presentation, we formalize them and prove that a logical syllogism with intermediate quantifiers is valid if it satisfies four extended Peterson’s rules.