Consider the following premises. Richard is not playing a game. For all x_7, if x_7 were an office clerk, then x_7 is playing a game. For all x_7, x_7 is an office clerk if and only if x_7 will play a game. Can we infer the following from them? For all x_7, x_7 is an office clerk if and only if x_7 will play a game.
That is one of the premises. Therefore, the answer is yes.