Consider the following premises. For all x_10, x_10 will drink water if and only if x_10 is a police officer. For all x_10, x_10 will listen to a song. For all x_10, x_10 will not play a game or x_10 will write a letter. For all x_10, x_10 will play a game or x_10 will drink water. For all x_10, x_10 is not a police officer. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_10 for which either x_10 will write a letter or x_10 will listen to a song are not true.
 From the fact that for all x_10, x_10 will play a game or x_10 will drink water, and that for all x_10, x_10 will not play a game or x_10 will write a letter we can infer that for all x_10, x_10 will drink water or x_10 will write a letter via universal resolution. From the fact that for all x_10, x_10 will drink water if and only if x_10 is a police officer, and that for all x_10, x_10 is not a police officer we can infer that for all x_10, x_10 will not drink water via universal biconditional elimination. From the fact that for all x_10, x_10 will drink water or x_10 will write a letter, and that for all x_10, x_10 will not drink water we can infer that for all x_10, x_10 will write a letter via universal disjunctive syllogism. Finally, from the fact that for all x_10, x_10 will write a letter, and that for all x_10, x_10 will listen to a song we can infer that for all x_10, x_10 will write a letter and x_10 will listen to a song via universal conjunction, which contradicts that There is at least one x_10 for which either x_10 will write a letter or x_10 will listen to a song are not true. Therefore, the answer is no.