Consider the following premises. For all x_10, if x_10 were a nurse, then x_10 is a janitor. For all x_10, if x_10 listens to a song, then x_10 is a nurse. For all x_10, x_10 will listen to a song if and only if x_10 is a janitor. For all x_10, if x_10 were a janitor, then x_10 is listening to a song. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_10 for which if is not true that x_10 will listen to a song if and only if x_10 is a janitor.
 From the fact that for all x_10, if x_10 listens to a song, then x_10 is a nurse, and that for all x_10, if x_10 were a nurse, then x_10 is a janitor we can infer that for all x_10, if x_10 listens to a song, then x_10 is a janitor via universal transitivity. From the fact that for all x_10, if x_10 were a janitor, then x_10 is listening to a song, and that for all x_10, if x_10 listens to a song, then x_10 is a janitor we can infer that for all x_10, x_10 is a janitor if and only if x_10 will listen to a song via universal biconditional introduction. From the fact that for all x_10, x_10 is a janitor if and only if x_10 will listen to a song we can infer that for all x_10, if x_10 listens to a song, then x_10 is a janitor via universal biconditional elimination. From the fact that for all x_10, x_10 will listen to a song if and only if x_10 is a janitor we can infer that for all x_10, if x_10 were a janitor, then x_10 is listening to a song via universal biconditional elimination. Finally, from the fact that for all x_10, if x_10 listens to a song, then x_10 is a janitor, and that for all x_10, if x_10 were a janitor, then x_10 is listening to a song we can infer that for all x_10, x_10 will listen to a song if and only if x_10 is a janitor via universal biconditional introduction, which contradicts that There is at least one x_10 for which if is not true that x_10 will listen to a song if and only if x_10 is a janitor. Therefore, the answer is no.