Consider the following premises. Patricia is cooking if and only if Karen is a janitor. If Barbara were happy, then Patricia is cooking. If Patricia cooks, then Barbara is happy. Can we infer the following from them? If we can, name the inference rule being used: The fact that Karen were a janitor does not imply that Barbara is happy.
 From the fact that Patricia is cooking if and only if Karen is a janitor we can infer that if Karen were a janitor, then Patricia is cooking via biconditional elimination. From the fact that if Patricia cooks, then Barbara is happy, and that if Barbara were happy, then Patricia is cooking we can infer that Patricia is cooking if and only if Barbara is happy via biconditional introduction. From the fact that Patricia is cooking if and only if Barbara is happy we can infer that if Patricia cooks, then Barbara is happy via biconditional elimination. Finally, from the fact that if Karen were a janitor, then Patricia is cooking, and that if Patricia cooks, then Barbara is happy we can infer that if Karen were a janitor, then Barbara is happy via transitivity, which contradicts that The fact that Karen were a janitor does not imply that Barbara is happy. Therefore, the answer is no.