Consider the following premises. If Karen goes running, then Barbara is cooking. If Karen goes running, then Barbara is cooking. If Barbara cooks, then Karen is running. Can we infer the following from them? Barbara is cooking if and only if Karen is running.
 From the fact that if Karen goes running, then Barbara is cooking, and that if Barbara cooks, then Karen is running we can infer that Karen is running if and only if Barbara is cooking. From the fact that Karen is running if and only if Barbara is cooking we can infer that if Barbara cooks, then Karen is running. Finally, from the fact that if Barbara cooks, then Karen is running, and that if Karen goes running, then Barbara is cooking we can infer that Barbara is cooking if and only if Karen is running. Therefore, the answer is yes.