Consider the following premises. For all y_4, y_4 is a lawyer or y_4 is not a politician. There is at least one y_4 for which y_4 will play a game or y_4 is a politician. For all y_4, y_4 will play a game if and only if y_4 is a lawyer. Can we infer the following from them? If we can, name the inference rule being used: There is at least one y_4 for which y_4 is a lawyer.
 From the fact that for all y_4, y_4 will play a game if and only if y_4 is a lawyer we can infer that for all y_4, if y_4 plays a game, then y_4 is a lawyer via universal biconditional elimination. From the fact that for all y_4, y_4 is a lawyer or y_4 is not a politician we can infer that for all y_4, if y_4 were a politician, then y_4 is a lawyer via universal rewriting. Finally, from the fact that for all y_4, if y_4 plays a game, then y_4 is a lawyer, that for all y_4, if y_4 were a politician, then y_4 is a lawyer, and that there is at least one y_4 for which y_4 will play a game or y_4 is a politician we can infer that there is at least one y_4 for which y_4 is a lawyer via existential disjunction elimination. Therefore, the answer is yes.