Consider the following premises. For all z, if z were sad, then z is a doctor. There is at least one z for which z will play a game if and only if z is sad. For all z, z is not a doctor. For all z, z will play a game or z is a police officer. Can we infer the following from them? If we can, name the inference rule being used: For all z, z is not a police officer.
 From the fact that for all z, if z were sad, then z is a doctor, and that for all z, z is not a doctor we can infer that for all z, z is not sad via universal modus tollens. From the fact that there is at least one z for which z will play a game if and only if z is sad, and that for all z, z is not sad we can infer that there is at least one z for which z will not play a game via existential biconditional elimination. Finally, from the fact that for all z, z will play a game or z is a police officer, and that there is at least one z for which z will not play a game we can infer that there is at least one z for which z is a police officer via existential disjunctive syllogism, which contradicts that For all z, z is not a police officer. Therefore, the answer is no.