Consider the following premises. There is at least one x_7 for which x_7 is curious if and only if x_7 is a janitor. For all x_7, x_7 will not listen to a song. For all x_7, x_7 will listen to a song or x_7 is not a janitor. For all x_7, if x_7 listens to a song, then x_7 is a janitor. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_7 for which x_7 is not curious.
 From the fact that for all x_7, x_7 will listen to a song or x_7 is not a janitor we can infer that for all x_7, if x_7 were a janitor, then x_7 is listening to a song via universal rewriting. From the fact that for all x_7, if x_7 listens to a song, then x_7 is a janitor, and that for all x_7, if x_7 were a janitor, then x_7 is listening to a song we can infer that for all x_7, x_7 will listen to a song if and only if x_7 is a janitor via universal biconditional introduction. From the fact that for all x_7, x_7 will listen to a song if and only if x_7 is a janitor, and that for all x_7, x_7 will not listen to a song we can infer that for all x_7, x_7 is not a janitor via universal biconditional elimination. Finally, from the fact that there is at least one x_7 for which x_7 is curious if and only if x_7 is a janitor, and that for all x_7, x_7 is not a janitor we can infer that there is at least one x_7 for which x_7 is not curious via existential biconditional elimination. Therefore, the answer is yes.