Consider the following premises. For all x_5, x_5 is a doctor and x_5 will climb a mountain. There is at least one x_5 for which if x_5 were a writer, then x_5 is playing tennis. For all x_5, if x_5 were a doctor, 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 for all x_5, x_5 is a doctor and x_5 will climb a mountain we can infer that for all x_5, x_5 is a doctor via universal simplification. From the fact that for all x_5, x_5 is a doctor we can infer that for all x_5, x_5 is a doctor or x_5 is a writer via universal addition. Finally, from the fact that for all x_5, if x_5 were a doctor, then x_5 is playing tennis, that there is at least one x_5 for which if x_5 were a writer, then x_5 is playing tennis, and that for all x_5, x_5 is a doctor or x_5 is a writer 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.