Consider the following premises. If William were an artist, then James is playing a game. For all x_15, x_15 will play a game if and only if x_15 will write a letter. If James writes a letter, then William is an artist. Can we infer the following from them? For all x_15, x_15 will play a game if and only if x_15 will write a letter.
That is one of the premises. Therefore, the answer is yes.