Consider the following premises. For all y_2, y_2 will play a game if and only if y_2 is an astronaut. There is at least one y_2 for which if y_2 were an astronaut, then y_2 is playing a game. For all y_2, if y_2 were an astronaut, then y_2 is playing a game. Can we infer the following from them? If we can, name the inference rule being used: For all y_2, the fact that y_2 plays a game does not imply that y_2 is an astronaut.
That contradicts the premises. Therefore the answer is no.