Consider the following premises. There is at least one y for which y will listen to a song if and only if y is a carpenter. For all y, y is not a doctor. For all y, if y were a carpenter, then y is a doctor. For all y, if y were a doctor, then y is a carpenter. 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 will not listen to a song.
 From the fact that for all y, if y were a doctor, then y is a carpenter, and that for all y, if y were a carpenter, then y is a doctor we can infer that for all y, y is a doctor if and only if y is a carpenter via universal biconditional introduction. From the fact that for all y, y is a doctor if and only if y is a carpenter, and that for all y, y is not a doctor we can infer that for all y, y is not a carpenter via universal biconditional elimination. Finally, from the fact that there is at least one y for which y will listen to a song if and only if y is a carpenter, and that for all y, y is not a carpenter we can infer that there is at least one y for which y will not listen to a song via existential biconditional elimination. Therefore, the answer is yes.