Consider the following premises. For all x_8, x_8 will cook if and only if x_8 will listen to a song. There is at least one x_8 for which if x_8 works, then x_8 is listening to a song. For all x_8, if x_8 reads a book, then x_8 is working. For all x_8, x_8 will cook or x_8 will read a book. 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 listen to a song.
 From the fact that for all x_8, x_8 will cook if and only if x_8 will listen to a song we can infer that for all x_8, if x_8 cooks, then x_8 is listening to a song via universal biconditional elimination. From the fact that for all x_8, if x_8 reads a book, then x_8 is working, and that there is at least one x_8 for which if x_8 works, then x_8 is listening to a song we can infer that there is at least one x_8 for which if x_8 reads a book, then x_8 is listening to a song via existential transitivity. Finally, from the fact that for all x_8, if x_8 cooks, then x_8 is listening to a song, that there is at least one x_8 for which if x_8 reads a book, then x_8 is listening to a song, and that for all x_8, x_8 will cook or x_8 will read a book we can infer that there is at least one x_8 for which x_8 will listen to a song via existential disjunction elimination. Therefore, the answer is yes.