Consider the following premises. For all x_10, x_10 is not a politician. For all x_10, if x_10 writes a letter, then x_10 is playing a game. There is at least one x_10 for which if x_10 were a politician, then x_10 is playing a game. For all x_10, if x_10 plays a game, then x_10 is a politician. Can we infer the following from them? If we can, name the inference rule being used: For all x_10, x_10 is not a politician.
That is one of the premises. Therefore, the answer is yes.