Consider the following premises. For all z, if z cooks, then z is bored. For all z, if z were bored, then z is sleeping. There is at least one z for which z is a mechanic or z will cook. For all z, z will play squash. For all z, if z were a mechanic, then z is sleeping. Can we infer the following from them? If we can, name the inference rule being used: For all z, either z will sleep or z will play squash are not true.
 From the fact that for all z, if z cooks, then z is bored, and that for all z, if z were bored, then z is sleeping we can infer that for all z, if z cooks, then z is sleeping via universal transitivity. From the fact that for all z, if z were a mechanic, then z is sleeping, that for all z, if z cooks, then z is sleeping, and that there is at least one z for which z is a mechanic or z will cook we can infer that there is at least one z for which z will sleep via existential disjunction elimination. Finally, from the fact that there is at least one z for which z will sleep, and that for all z, z will play squash we can infer that there is at least one z for which z will sleep and z will play squash via existential conjunction, which contradicts that For all z, either z will sleep or z will play squash are not true. Therefore, the answer is no.