Consider the following premises. For all x_8, x_8 is not bored. There is at least one x_8 for which x_8 will cook or x_8 will not listen to a song. For all x_8, if x_8 listens to a song, then x_8 is bored. For all x_8, if x_8 cooks, then x_8 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_8 for which x_8 will not cook.
 From the fact that for all x_8, if x_8 listens to a song, then x_8 is bored, and that for all x_8, x_8 is not bored we can infer that for all x_8, x_8 will not listen to a song via universal modus tollens. From the fact that there is at least one x_8 for which x_8 will cook or x_8 will not listen to a song we can infer that there is at least one x_8 for which if x_8 listens to a song, then x_8 is cooking via existential rewriting. From the fact that for all x_8, if x_8 cooks, then x_8 is listening to a song, and that there is at least one x_8 for which if x_8 listens to a song, then x_8 is cooking we can infer that there is at least one x_8 for which x_8 will cook if and only if x_8 will listen to a song via existential biconditional introduction. Finally, from the fact that there is at least one x_8 for which x_8 will cook if and only if x_8 will listen to a song, and that for all x_8, x_8 will not listen to a song we can infer that there is at least one x_8 for which x_8 will not cook via existential biconditional elimination. Therefore, the answer is yes.