Consider the following premises. For all x_8, if x_8 were poor, then x_8 is cooking. There is at least one x_8 for which if x_8 were a cashier, then x_8 is cooking. For all x_8, x_8 will listen to a song. For all x_8, x_8 is poor or x_8 is a cashier. Can we infer the following from them? If we can, name the inference rule being used: For all x_8, either x_8 will cook or x_8 will listen to a song are not true.
 From the fact that for all x_8, if x_8 were poor, then x_8 is cooking, that there is at least one x_8 for which if x_8 were a cashier, then x_8 is cooking, and that for all x_8, x_8 is poor or x_8 is a cashier we can infer that there is at least one x_8 for which x_8 will cook via existential disjunction elimination. Finally, from the fact that there is at least one x_8 for which x_8 will cook, and that for all x_8, x_8 will listen to a song we can infer that there is at least one x_8 for which x_8 will cook and x_8 will listen to a song via existential conjunction, which contradicts that For all x_8, either x_8 will cook or x_8 will listen to a song are not true. Therefore, the answer is no.