Consider the following premises. There is at least one y for which if y listens to a song, then y is writing a letter. For all y, y will listen to a song. For all y, if y were a nurse, then y is not writing a letter. Can we infer the following from them? If we can, name the inference rule being used: There is at least one y for which y is not a nurse.
 From the fact that there is at least one y for which if y listens to a song, then y is writing a letter, and that for all y, y will listen to a song we can infer that there is at least one y for which y will write a letter via existential modus ponens. Finally, from the fact that for all y, if y were a nurse, then y is not writing a letter, and that there is at least one y for which y will write a letter we can infer that there is at least one y for which y is not a nurse via existential modus tollens. Therefore, the answer is yes.