Consider the following premises. For all x_5, x_5 will go running. For all x_5, x_5 will play a game or x_5 will not write a letter. There is at least one x_5 for which if x_5 goes running, then x_5 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_5 for which x_5 will play a game.
 From the fact that for all x_5, x_5 will go running we can infer that for all x_5, x_5 will go running or x_5 will write a letter via universal addition. From the fact that for all x_5, x_5 will play a game or x_5 will not write a letter we can infer that for all x_5, if x_5 writes a letter, then x_5 is playing a game via universal rewriting. Finally, from the fact that there is at least one x_5 for which if x_5 goes running, then x_5 is playing a game, that for all x_5, if x_5 writes a letter, then x_5 is playing a game, and that for all x_5, x_5 will go running or x_5 will write a letter we can infer that there is at least one x_5 for which x_5 will play a game via existential disjunction elimination. Therefore, the answer is yes.