Consider the following premises. For all x_10, x_10 is a cashier if and only if x_10 is a soldier. For all x_10, x_10 is a cashier or x_10 will not climb a mountain. For all x_10, x_10 will climb a mountain or x_10 is a soldier. 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 neither x_10 is a cashier nor x_10 will listen to a song.
 From the fact that for all x_10, x_10 is a cashier or x_10 will not climb a mountain we can infer that for all x_10, if x_10 climbs a mountain, then x_10 is a cashier via universal rewriting. From the fact that for all x_10, x_10 is a cashier if and only if x_10 is a soldier we can infer that for all x_10, if x_10 were a soldier, then x_10 is a cashier via universal biconditional elimination. From the fact that for all x_10, if x_10 climbs a mountain, then x_10 is a cashier, that for all x_10, if x_10 were a soldier, then x_10 is a cashier, and that for all x_10, x_10 will climb a mountain or x_10 is a soldier we can infer that for all x_10, x_10 is a cashier via universal disjunction elimination. Finally, from the fact that for all x_10, x_10 is a cashier we can infer that for all x_10, x_10 is a cashier or x_10 will listen to a song via universal addition, which contradicts that There is at least one x_10 for which neither x_10 is a cashier nor x_10 will listen to a song. Therefore, the answer is no.