Consider the following premises. It is overcast if and only if Mary is a police officer. John is not cooking. John is cooking if and only if Mary is a scientist. If it is overcast, then Mary is a scientist. Can we infer the following from them? Mary is not a police officer.
 From the fact that John is cooking if and only if Mary is a scientist, and that John is not cooking we can infer that Mary is not a scientist. From the fact that if it is overcast, then Mary is a scientist, and that Mary is not a scientist we can infer that it is not overcast. Finally, from the fact that it is overcast if and only if Mary is a police officer, and that it is not overcast we can infer that Mary is not a police officer. Therefore, the answer is yes.