Consider the following premises. There is at least one z for which if z plays squash, then z is playing tennis. For all z, if z plays tennis, then z is a cashier. For all z, z will play squash or z is not a janitor. Can we infer the following from them? If we can, name the inference rule being used: There is at least one z for which if z were a janitor, then z is a cashier.
 From the fact that for all z, z will play squash or z is not a janitor we can infer that for all z, if z were a janitor, then z is playing squash via universal rewriting. From the fact that there is at least one z for which if z plays squash, then z is playing tennis, and that for all z, if z plays tennis, then z is a cashier we can infer that there is at least one z for which if z plays squash, then z is a cashier via existential transitivity. Finally, from the fact that for all z, if z were a janitor, then z is playing squash, and that there is at least one z for which if z plays squash, then z is a cashier we can infer that there is at least one z for which if z were a janitor, then z is a cashier via existential transitivity. Therefore, the answer is yes.