Consider the following premises. There is at least one x for which x will drink water. For all x, if x goes running, then x is a janitor. For all x, x will go running. For all x, if x were a janitor, then x is cooking. Can we infer the following from them? If we can, name the inference rule being used: For all x, either x will cook or x will drink water are not true.
 From the fact that for all x, if x goes running, then x is a janitor, and that for all x, if x were a janitor, then x is cooking we can infer that for all x, if x goes running, then x is cooking via universal transitivity. From the fact that for all x, if x goes running, then x is cooking, and that for all x, x will go running we can infer that for all x, x will cook via universal modus ponens. Finally, from the fact that for all x, x will cook, and that there is at least one x for which x will drink water we can infer that there is at least one x for which x will cook and x will drink water via existential conjunction, which contradicts that For all x, either x will cook or x will drink water are not true. Therefore, the answer is no.