Consider the following premises. There is at least one x for which x will work or x is happy. For all x, x is a cashier if and only if x will work. For all x, x is not a cashier. For all x, if x works, then x is a cashier. Can we infer the following from them? If we can, name the inference rule being used: For all x, x is not happy.
 From the fact that for all x, x is a cashier if and only if x will work we can infer that for all x, if x were a cashier, then x is working via universal biconditional elimination. From the fact that for all x, if x were a cashier, then x is working, and that for all x, if x works, then x is a cashier we can infer that for all x, x is a cashier if and only if x will work via universal biconditional introduction. From the fact that for all x, x is a cashier if and only if x will work, and that for all x, x is not a cashier we can infer that for all x, x will not work via universal biconditional elimination. Finally, from the fact that there is at least one x for which x will work or x is happy, and that for all x, x will not work we can infer that there is at least one x for which x is happy via existential disjunctive syllogism, which contradicts that For all x, x is not happy. Therefore, the answer is no.