Consider the following premises. There is at least one z for which if z goes running, then z is a mechanic. For all z, if z were a mechanic, then z is running. For all z, if z sleeps, then z is running. 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 sleeps, then z is a mechanic.
 From the fact that for all z, if z were a mechanic, then z is running, and that there is at least one z for which if z goes running, then z is a mechanic we can infer that there is at least one z for which z is a mechanic if and only if z will go running via existential biconditional introduction. From the fact that there is at least one z for which z is a mechanic if and only if z will go running we can infer that there is at least one z for which if z goes running, then z is a mechanic via existential biconditional elimination. Finally, from the fact that for all z, if z sleeps, then z is running, and that there is at least one z for which if z goes running, then z is a mechanic we can infer that there is at least one z for which if z sleeps, then z is a mechanic via existential transitivity. Therefore, the answer is yes.