Consider the following premises. There is at least one z for which if z were a bartender, then z is bored. For all z, if z were a doctor, then z is a bartender. For all z, if z were bored, then z is a doctor. For all z, z is not a bartender. Can we infer the following from them? If we can, name the inference rule being used: For all z, z is a doctor.
 From the fact that there is at least one z for which if z were a bartender, then z is bored, and that for all z, if z were bored, then z is a doctor we can infer that there is at least one z for which if z were a bartender, then z is a doctor via existential transitivity. From the fact that for all z, if z were a doctor, then z is a bartender, and that there is at least one z for which if z were a bartender, then z is a doctor we can infer that there is at least one z for which z is a doctor if and only if z is a bartender via existential biconditional introduction. Finally, from the fact that there is at least one z for which z is a doctor if and only if z is a bartender, and that for all z, z is not a bartender we can infer that there is at least one z for which z is not a doctor via existential biconditional elimination, which contradicts that For all z, z is a doctor. Therefore, the answer is no.