Consider the following premises. If it is cloudy, then Barbara is a soldier. It is cloudy or Barbara is not a soldier. Barbara is not a soldier. Can we infer the following from them? If we can, name the inference rule being used: It is not cloudy.
 From the fact that it is cloudy or Barbara is not a soldier we can infer that if Barbara were a soldier, then it is cloudy via rewriting. From the fact that if it is cloudy, then Barbara is a soldier, and that if Barbara were a soldier, then it is cloudy we can infer that it is cloudy if and only if Barbara is a soldier via biconditional introduction. Finally, from the fact that it is cloudy if and only if Barbara is a soldier, and that Barbara is not a soldier we can infer that it is not cloudy via biconditional elimination. Therefore, the answer is yes.