Consider the following premises. For all x_5, if x_5 were an office clerk, then x_5 is cooking. For all x_5, if x_5 cooks, then x_5 is a lawyer. For all x_5, x_5 is bored or x_5 is an office clerk. There is at least one x_5 for which if x_5 plays tennis, then x_5 is not a lawyer. For all x_5, if x_5 were bored, then x_5 is a lawyer. Can we infer the following from them? If we can, name the inference rule being used: For all x_5, x_5 will play tennis.
 From the fact that for all x_5, if x_5 were an office clerk, then x_5 is cooking, and that for all x_5, if x_5 cooks, then x_5 is a lawyer we can infer that for all x_5, if x_5 were an office clerk, then x_5 is a lawyer via universal transitivity. From the fact that for all x_5, if x_5 were bored, then x_5 is a lawyer, that for all x_5, if x_5 were an office clerk, then x_5 is a lawyer, and that for all x_5, x_5 is bored or x_5 is an office clerk we can infer that for all x_5, x_5 is a lawyer via universal disjunction elimination. Finally, from the fact that there is at least one x_5 for which if x_5 plays tennis, then x_5 is not a lawyer, and that for all x_5, x_5 is a lawyer we can infer that there is at least one x_5 for which x_5 will not play tennis via existential modus tollens, which contradicts that For all x_5, x_5 will play tennis. Therefore, the answer is no.