Consider the following premises. For all z, if z plays tennis, then z is a lawyer. There is at least one z for which if z were a lawyer, then z is playing tennis. For all z, if z climbs a mountain, then z is a lawyer. For all z, z will play tennis. Can we infer the following from them? If we can, name the inference rule being used: There is at least one z for which z is a lawyer.
 From the fact that for all z, if z plays tennis, then z is a lawyer, and that there is at least one z for which if z were a lawyer, then z is playing tennis we can infer that there is at least one z for which z will play tennis if and only if z is a lawyer via existential biconditional introduction. From the fact that there is at least one z for which z will play tennis if and only if z is a lawyer we can infer that there is at least one z for which if z plays tennis, then z is a lawyer via existential biconditional elimination. From the fact that for all z, z will play tennis we can infer that for all z, z will play tennis or z will climb a mountain via universal addition. Finally, from the fact that there is at least one z for which if z plays tennis, then z is a lawyer, that for all z, if z climbs a mountain, then z is a lawyer, and that for all z, z will play tennis or z will climb a mountain we can infer that there is at least one z for which z is a lawyer via existential disjunction elimination. Therefore, the answer is yes.