Consider the following premises. John is a writer if and only if Robert is playing a game. For all x_10, if x_10 were a writer, then x_10 is climbing a mountain. John is not climbing a mountain. Can we infer the following from them? Robert is not playing a game.
 From the fact that for all x_10, if x_10 were a writer, then x_10 is climbing a mountain we can infer that if John were a writer, then John is climbing a mountain. From the fact that if John were a writer, then John is climbing a mountain, and that John is not climbing a mountain we can infer that John is not a writer. Finally, from the fact that John is a writer if and only if Robert is playing a game, and that John is not a writer we can infer that Robert is not playing a game. Therefore, the answer is yes.