Consider the following premises. For all x_4, x_4 is rich or x_4 is a carpenter. There is at least one x_4 for which if x_4 plays a game, then x_4 is a politician. For all x_4, if x_4 were a carpenter, then x_4 is a politician. For all x_4, if x_4 were a mechanic, then x_4 is playing a game. For all x_4, x_4 is not rich or x_4 is a mechanic. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_4 for which x_4 is a politician.
 From the fact that for all x_4, if x_4 were a mechanic, then x_4 is playing a game, and that there is at least one x_4 for which if x_4 plays a game, then x_4 is a politician we can infer that there is at least one x_4 for which if x_4 were a mechanic, then x_4 is a politician via existential transitivity. From the fact that for all x_4, x_4 is rich or x_4 is a carpenter, and that for all x_4, x_4 is not rich or x_4 is a mechanic we can infer that for all x_4, x_4 is a carpenter or x_4 is a mechanic via universal resolution. Finally, from the fact that for all x_4, if x_4 were a carpenter, then x_4 is a politician, that there is at least one x_4 for which if x_4 were a mechanic, then x_4 is a politician, and that for all x_4, x_4 is a carpenter or x_4 is a mechanic we can infer that there is at least one x_4 for which x_4 is a politician via existential disjunction elimination. Therefore, the answer is yes.