Consider the following premises. If William were a nurse, then Barbara is listening to a song. For all y_2, y_2 will not listen to a song. If Barbara listens to a song, then William is a nurse. Can we infer the following from them? If we can, name the inference rule being used: William is not a nurse.
 From the fact that for all y_2, y_2 will not listen to a song we can infer that Barbara is not listening to a song via universal instantiation. From the fact that if William were a nurse, then Barbara is listening to a song, and that if Barbara listens to a song, then William is a nurse we can infer that William is a nurse if and only if Barbara is listening to a song via biconditional introduction. Finally, from the fact that William is a nurse if and only if Barbara is listening to a song, and that Barbara is not listening to a song we can infer that William is not a nurse via biconditional elimination. Therefore, the answer is yes.