Consider the following premises. For all y, if y were a scientist, then y is playing tennis. There is at least one y for which y is a scientist if and only if y is a lawyer. For all y, y will not play tennis. Can we infer the following from them? If we can, name the inference rule being used: There is at least one y for which y is not a lawyer.
 From the fact that there is at least one y for which y is a scientist if and only if y is a lawyer we can infer that there is at least one y for which if y were a lawyer, then y is a scientist via existential biconditional elimination. From the fact that there is at least one y for which if y were a lawyer, then y is a scientist, and that for all y, if y were a scientist, then y is playing tennis we can infer that there is at least one y for which if y were a lawyer, then y is playing tennis via existential transitivity. Finally, from the fact that there is at least one y for which if y were a lawyer, then y is playing tennis, and that for all y, y will not play tennis we can infer that there is at least one y for which y is not a lawyer via existential modus tollens. Therefore, the answer is yes.