Consider the following premises. For all x_5, if x_5 goes running, then x_5 is cooking. For all x_5, x_5 will take a plane or x_5 will go running. There is at least one x_5 for which if x_5 takes a plane, then x_5 is a cashier. For all x_5, if x_5 were a cashier, then x_5 is playing tennis. For all x_5, if x_5 cooks, then x_5 is playing tennis. Can we infer the following from them? If we can, name the inference rule being used: For all x_5, x_5 will not play tennis.
 From the fact that there is at least one x_5 for which if x_5 takes a plane, then x_5 is a cashier, and that for all x_5, if x_5 were a cashier, then x_5 is playing tennis we can infer that there is at least one x_5 for which if x_5 takes a plane, then x_5 is playing tennis via existential transitivity. From the fact that for all x_5, if x_5 goes running, then x_5 is cooking, and that for all x_5, if x_5 cooks, then x_5 is playing tennis we can infer that for all x_5, if x_5 goes running, then x_5 is playing tennis via universal transitivity. Finally, from the fact that there is at least one x_5 for which if x_5 takes a plane, then x_5 is playing tennis, that for all x_5, if x_5 goes running, then x_5 is playing tennis, and that for all x_5, x_5 will take a plane or x_5 will go running we can infer that there is at least one x_5 for which x_5 will play tennis via existential disjunction elimination, which contradicts that For all x_5, x_5 will not play tennis. Therefore, the answer is no.