Consider the following premises. There is at least one x_2 for which if x_2 were a poet, then x_2 is playing a game. For all x_2, if x_2 were a carpenter, then x_2 is a poet. For all x_2, x_2 is a carpenter or x_2 is a doctor. For all x_2, if x_2 were a doctor, then x_2 is playing a game. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_2 for which x_2 will play a game.
 From the fact that for all x_2, if x_2 were a carpenter, then x_2 is a poet, and that there is at least one x_2 for which if x_2 were a poet, then x_2 is playing a game we can infer that there is at least one x_2 for which if x_2 were a carpenter, then x_2 is playing a game via existential transitivity. Finally, from the fact that there is at least one x_2 for which if x_2 were a carpenter, then x_2 is playing a game, that for all x_2, if x_2 were a doctor, then x_2 is playing a game, and that for all x_2, x_2 is a carpenter or x_2 is a doctor we can infer that there is at least one x_2 for which x_2 will play a game via existential disjunction elimination. Therefore, the answer is yes.