Consider the following premises. For all x_1, x_1 will play squash if and only if x_1 is a lawyer. For all x_1, if x_1 plays squash, then x_1 is a lawyer. For all x_1, if x_1 were a lawyer, then x_1 is playing squash. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_1 for which if is not true that x_1 will play squash if and only if x_1 is a lawyer.
 From the fact that for all x_1, if x_1 plays squash, then x_1 is a lawyer, and that for all x_1, if x_1 were a lawyer, then x_1 is playing squash we can infer that for all x_1, x_1 will play squash if and only if x_1 is a lawyer via universal biconditional introduction. From the fact that for all x_1, x_1 will play squash if and only if x_1 is a lawyer we can infer that for all x_1, if x_1 were a lawyer, then x_1 is playing squash via universal biconditional elimination. From the fact that for all x_1, x_1 will play squash if and only if x_1 is a lawyer we can infer that for all x_1, if x_1 plays squash, then x_1 is a lawyer via universal biconditional elimination. Finally, from the fact that for all x_1, if x_1 plays squash, then x_1 is a lawyer, and that for all x_1, if x_1 were a lawyer, then x_1 is playing squash we can infer that for all x_1, x_1 will play squash if and only if x_1 is a lawyer via universal biconditional introduction, which contradicts that There is at least one x_1 for which if is not true that x_1 will play squash if and only if x_1 is a lawyer. Therefore, the answer is no.