Consider the following premises. For all x_5, x_5 is a nurse if and only if x_5 will go running. For all x_5, if x_5 goes running, then x_5 is a poet. For all x_5, if x_5 were a poet, then x_5 is a nurse. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_5 for which if is not true that x_5 is a nurse if and only if x_5 will go running.
 From the fact that for all x_5, if x_5 goes running, then x_5 is a poet, and that for all x_5, if x_5 were a poet, then x_5 is a nurse we can infer that for all x_5, if x_5 goes running, then x_5 is a nurse via universal transitivity. From the fact that for all x_5, x_5 is a nurse if and only if x_5 will go running we can infer that for all x_5, if x_5 were a nurse, then x_5 is running via universal biconditional elimination. Finally, from the fact that for all x_5, if x_5 were a nurse, then x_5 is running, and that for all x_5, if x_5 goes running, then x_5 is a nurse we can infer that for all x_5, x_5 is a nurse if and only if x_5 will go running via universal biconditional introduction, which contradicts that There is at least one x_5 for which if is not true that x_5 is a nurse if and only if x_5 will go running. Therefore, the answer is no.