Consider the following premises. There is at least one x_13 for which x_13 is a janitor or x_13 will write a letter. For all x_13, x_13 is a janitor if and only if x_13 will go running. For all x_13, x_13 will not go running. Can we infer the following from them? If we can, name the inference rule being used: For all x_13, neither x_13 will write a letter nor x_13 will listen to a song.
 From the fact that for all x_13, x_13 is a janitor if and only if x_13 will go running, and that for all x_13, x_13 will not go running we can infer that for all x_13, x_13 is not a janitor via universal biconditional elimination. From the fact that there is at least one x_13 for which x_13 is a janitor or x_13 will write a letter, and that for all x_13, x_13 is not a janitor we can infer that there is at least one x_13 for which x_13 will write a letter via existential disjunctive syllogism. Finally, from the fact that there is at least one x_13 for which x_13 will write a letter we can infer that there is at least one x_13 for which x_13 will write a letter or x_13 will listen to a song via existential addition, which contradicts that For all x_13, neither x_13 will write a letter nor x_13 will listen to a song. Therefore, the answer is no.