Consider the following premises. For all x_3, x_3 is a taxi driver or x_3 will not play a game. There is at least one x_3 for which x_3 will play a game or x_3 is not excited. 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 if x_3 were excited, then x_3 is a taxi driver.
 From the fact that there is at least one x_3 for which x_3 will play a game or x_3 is not excited we can infer that there is at least one x_3 for which if x_3 were excited, then x_3 is playing a game via existential rewriting. From the fact that for all x_3, x_3 is a taxi driver or x_3 will not play a game we can infer that for all x_3, if x_3 plays a game, then x_3 is a taxi driver via universal rewriting. Finally, from the fact that there is at least one x_3 for which if x_3 were excited, then x_3 is playing a game, and that for all x_3, if x_3 plays a game, then x_3 is a taxi driver we can infer that there is at least one x_3 for which if x_3 were excited, then x_3 is a taxi driver via existential transitivity. Therefore, the answer is yes.