Consider the following premises. There is at least one x_16 for which x_16 is a nurse if and only if x_16 will sleep. For all x_16, if x_16 sleeps, then x_16 is playing a game. For all x_16, x_16 will not go running. For all x_16, x_16 will go running if and only if x_16 is a poet. For all x_16, x_16 is a poet if and only if x_16 will play a game. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_16 for which x_16 is not a nurse.
 From the fact that for all x_16, x_16 will go running if and only if x_16 is a poet, and that for all x_16, x_16 will not go running we can infer that for all x_16, x_16 is not a poet via universal biconditional elimination. From the fact that for all x_16, x_16 is a poet if and only if x_16 will play a game, and that for all x_16, x_16 is not a poet we can infer that for all x_16, x_16 will not play a game via universal biconditional elimination. From the fact that for all x_16, if x_16 sleeps, then x_16 is playing a game, and that for all x_16, x_16 will not play a game we can infer that for all x_16, x_16 will not sleep via universal modus tollens. From the fact that there is at least one x_16 for which x_16 is a nurse if and only if x_16 will sleep we can infer that there is at least one x_16 for which if x_16 were a nurse, then x_16 is sleeping via existential biconditional elimination. Finally, from the fact that there is at least one x_16 for which if x_16 were a nurse, then x_16 is sleeping, and that for all x_16, x_16 will not sleep we can infer that there is at least one x_16 for which x_16 is not a nurse via existential modus tollens. Therefore, the answer is yes.