Consider the following premises. For all x_3, if x_3 were an astronaut, then x_3 is a doctor. There is at least one x_3 for which x_3 is a doctor or x_3 will not play a game. For all x_3, x_3 will play a game or x_3 is an astronaut. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_3 for which x_3 is a doctor.
 From the fact that there is at least one x_3 for which x_3 is a doctor or x_3 will not play a game we can infer that there is at least one x_3 for which if x_3 plays a game, then x_3 is a doctor via existential rewriting. Finally, from the fact that there is at least one x_3 for which if x_3 plays a game, then x_3 is a doctor, that for all x_3, if x_3 were an astronaut, then x_3 is a doctor, and that for all x_3, x_3 will play a game or x_3 is an astronaut we can infer that there is at least one x_3 for which x_3 is a doctor via existential disjunction elimination. Therefore, the answer is yes.