Consider the following premises. For all x_5, if x_5 were a poet, then x_5 is playing tennis. For all x_5, if x_5 reads a book, then x_5 is playing tennis. For all x_5, x_5 will read a book or x_5 is a cashier. For all x_5, if x_5 were a cashier, then x_5 is a poet. There is at least one x_5 for which x_5 is a nurse. Can we infer the following from them? If we can, name the inference rule being used: For all x_5, either x_5 will play tennis or x_5 is a nurse are not true.
 From the fact that for all x_5, if x_5 were a cashier, then x_5 is a poet, and that for all x_5, if x_5 were a poet, then x_5 is playing tennis we can infer that for all x_5, if x_5 were a cashier, then x_5 is playing tennis via universal transitivity. From the fact that for all x_5, if x_5 reads a book, then x_5 is playing tennis, that for all x_5, if x_5 were a cashier, then x_5 is playing tennis, and that for all x_5, x_5 will read a book or x_5 is a cashier we can infer that for all x_5, x_5 will play tennis via universal disjunction elimination. Finally, from the fact that for all x_5, x_5 will play tennis, and that there is at least one x_5 for which x_5 is a nurse we can infer that there is at least one x_5 for which x_5 will play tennis and x_5 is a nurse via existential conjunction, which contradicts that For all x_5, either x_5 will play tennis or x_5 is a nurse are not true. Therefore, the answer is no.