Consider the following premises. There is at least one x_18 for which x_18 is not a nurse. For all x_18, x_18 is a nurse or x_18 will write a letter. For all x_18, if x_18 were excited, then x_18 is a cashier. For all x_18, if x_18 were a cashier, then x_18 is running. For all x_18, if x_18 goes running, then x_18 is a cashier. For all x_18, x_18 is excited or x_18 will go running. Can we infer the following from them? If we can, name the inference rule being used: For all x_18, either x_18 is a cashier or x_18 will write a letter are not true.
 From the fact that for all x_18, if x_18 goes running, then x_18 is a cashier, and that for all x_18, if x_18 were a cashier, then x_18 is running we can infer that for all x_18, x_18 will go running if and only if x_18 is a cashier via universal biconditional introduction. From the fact that for all x_18, x_18 will go running if and only if x_18 is a cashier we can infer that for all x_18, if x_18 goes running, then x_18 is a cashier via universal biconditional elimination. From the fact that for all x_18, x_18 is a nurse or x_18 will write a letter, and that there is at least one x_18 for which x_18 is not a nurse we can infer that there is at least one x_18 for which x_18 will write a letter via existential disjunctive syllogism. From the fact that for all x_18, if x_18 were excited, then x_18 is a cashier, that for all x_18, if x_18 goes running, then x_18 is a cashier, and that for all x_18, x_18 is excited or x_18 will go running we can infer that for all x_18, x_18 is a cashier via universal disjunction elimination. Finally, from the fact that for all x_18, x_18 is a cashier, and that there is at least one x_18 for which x_18 will write a letter we can infer that some cashiers will write a letter via existential conjunction, which contradicts that For all x_18, either x_18 is a cashier or x_18 will write a letter are not true. Therefore, the answer is no.