Consider the following premises. There is at least one x_7 for which if x_7 were sad, then x_7 is a doctor. For all x_7, x_7 is not a doctor. For all x_7, if x_7 plays a game, then x_7 is sad. For all x_7, if x_7 plays tennis, then x_7 is playing a game. Can we infer the following from them? If we can, name the inference rule being used: For all x_7, x_7 will play tennis.
 From the fact that for all x_7, if x_7 plays a game, then x_7 is sad, and that there is at least one x_7 for which if x_7 were sad, then x_7 is a doctor we can infer that there is at least one x_7 for which if x_7 plays a game, then x_7 is a doctor via existential transitivity. From the fact that for all x_7, if x_7 plays tennis, then x_7 is playing a game, and that there is at least one x_7 for which if x_7 plays a game, then x_7 is a doctor we can infer that there is at least one x_7 for which if x_7 plays tennis, then x_7 is a doctor via existential transitivity. Finally, from the fact that there is at least one x_7 for which if x_7 plays tennis, then x_7 is a doctor, and that for all x_7, x_7 is not a doctor we can infer that there is at least one x_7 for which x_7 will not play tennis via existential modus tollens, which contradicts that For all x_7, x_7 will play tennis. Therefore, the answer is no.