Consider the following premises. If Mary were a soldier, then it is windy. It is raining or it is cloudy. If it rains, then John is cooking. If it is cloudy, then John is cooking. It is not windy. Can we infer the following from them? If we can, name the inference rule being used: Mary is a soldier.
 From the fact that if it rains, then John is cooking, that if it is cloudy, then John is cooking, and that it is raining or it is cloudy we can infer that John is cooking via disjunction elimination. Finally, from the fact that if Mary were a soldier, then it is windy, and that it is not windy we can infer that Mary is not a soldier via modus tollens, which contradicts that Mary is a soldier. Therefore, the answer is no.