Consider the following premises. For all x_2, if x_2 were an office clerk, then x_2 is playing squash. There is at least one x_2 for which x_2 will not play squash. For all x_2, if x_2 were a doctor, then x_2 is an office clerk. For all x_2, if x_2 plays squash, then x_2 is a doctor. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_2 for which x_2 is not an office clerk.
 From the fact that for all x_2, if x_2 plays squash, then x_2 is a doctor, and that for all x_2, if x_2 were a doctor, then x_2 is an office clerk we can infer that for all x_2, if x_2 plays squash, then x_2 is an office clerk via universal transitivity. From the fact that for all x_2, if x_2 plays squash, then x_2 is an office clerk, and that for all x_2, if x_2 were an office clerk, then x_2 is playing squash we can infer that for all x_2, x_2 will play squash if and only if x_2 is an office clerk via universal biconditional introduction. Finally, from the fact that for all x_2, x_2 will play squash if and only if x_2 is an office clerk, and that there is at least one x_2 for which x_2 will not play squash we can infer that there is at least one x_2 for which x_2 is not an office clerk via existential biconditional elimination. Therefore, the answer is yes.