Consider the following premises. For all x_4, x_4 is poor. If Joseph were a nurse, then Robert is playing a game. If Joseph were poor, then Robert is playing a game. Can we infer the following from them? If we can, name the inference rule being used: Robert is not playing a game.
 From the fact that for all x_4, x_4 is poor we can infer that for all x_4, x_4 is poor or x_4 is a nurse via universal addition. From the fact that for all x_4, x_4 is poor or x_4 is a nurse we can infer that Joseph is poor or Joseph is a nurse via universal instantiation. Finally, from the fact that if Joseph were poor, then Robert is playing a game, that if Joseph were a nurse, then Robert is playing a game, and that Joseph is poor or Joseph is a nurse we can infer that Robert is playing a game via disjunction elimination, which contradicts that Robert is not playing a game. Therefore, the answer is no.