Consider the following premises. There is at least one x_4 for which x_4 is not poor. For all x_4, x_4 will play a game if and only if x_4 is poor. For all x_4, if x_4 plays a game, then x_4 is climbing a mountain. For all x_4, if x_4 climbs a mountain, then x_4 is playing a game. Can we infer the following from them? For all x_4, x_4 will play a game if and only if x_4 is poor.
That is one of the premises. Therefore, the answer is yes.