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