Consider the following premises. There is at least one x_11 for which x_11 will write a letter if and only if x_11 will listen to a song. For all x_11, x_11 is not a doctor. For all x_11, x_11 will listen to a song if and only if x_11 will play a game. For all x_11, x_11 is a doctor if and only if x_11 will play a game. Can we infer the following from them? If we can, name the inference rule being used: For all x_11, x_11 will write a letter.
 From the fact that for all x_11, x_11 will listen to a song if and only if x_11 will play a game we can infer that for all x_11, if x_11 listens to a song, then x_11 is playing a game via universal biconditional elimination. From the fact that for all x_11, x_11 is a doctor if and only if x_11 will play a game, and that for all x_11, x_11 is not a doctor we can infer that for all x_11, x_11 will not play a game via universal biconditional elimination. From the fact that for all x_11, if x_11 listens to a song, then x_11 is playing a game, and that for all x_11, x_11 will not play a game we can infer that for all x_11, x_11 will not listen to a song via universal modus tollens. Finally, from the fact that there is at least one x_11 for which x_11 will write a letter if and only if x_11 will listen to a song, and that for all x_11, x_11 will not listen to a song we can infer that there is at least one x_11 for which x_11 will not write a letter via existential biconditional elimination, which contradicts that For all x_11, x_11 will write a letter. Therefore, the answer is no.