Consider the following premises. For all x_7, x_7 will cook if and only if x_7 is bored. There is at least one x_7 for which x_7 will cook or x_7 is not a cashier. For all x_7, x_7 is a cashier or x_7 is bored. Can we infer the following from them? If we can, name the inference rule being used: For all x_7, x_7 will not cook.
 From the fact that there is at least one x_7 for which x_7 will cook or x_7 is not a cashier we can infer that there is at least one x_7 for which if x_7 were a cashier, then x_7 is cooking via existential rewriting. From the fact that for all x_7, x_7 will cook if and only if x_7 is bored we can infer that for all x_7, if x_7 were bored, then x_7 is cooking via universal biconditional elimination. Finally, from the fact that there is at least one x_7 for which if x_7 were a cashier, then x_7 is cooking, that for all x_7, if x_7 were bored, then x_7 is cooking, and that for all x_7, x_7 is a cashier or x_7 is bored we can infer that there is at least one x_7 for which x_7 will cook via existential disjunction elimination, which contradicts that For all x_7, x_7 will not cook. Therefore, the answer is no.