Consider the following premises. For all x_3, x_3 is not an office clerk. There is at least one x_3 for which if x_3 sleeps, then x_3 is an office clerk. For all x_3, if x_3 were an office clerk, then x_3 is sleeping. For all x_3, x_3 is an office clerk or x_3 will not sleep. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_3 for which x_3 will not sleep.
 From the fact that there is at least one x_3 for which if x_3 sleeps, then x_3 is an office clerk, and that for all x_3, if x_3 were an office clerk, then x_3 is sleeping we can infer that there is at least one x_3 for which x_3 will sleep if and only if x_3 is an office clerk via existential biconditional introduction. From the fact that for all x_3, x_3 is an office clerk or x_3 will not sleep we can infer that for all x_3, if x_3 sleeps, then x_3 is an office clerk via universal rewriting. From the fact that there is at least one x_3 for which x_3 will sleep if and only if x_3 is an office clerk we can infer that there is at least one x_3 for which if x_3 were an office clerk, then x_3 is sleeping via existential biconditional elimination. From the fact that there is at least one x_3 for which if x_3 were an office clerk, then x_3 is sleeping, and that for all x_3, if x_3 sleeps, then x_3 is an office clerk we can infer that there is at least one x_3 for which x_3 is an office clerk if and only if x_3 will sleep via existential biconditional introduction. Finally, from the fact that there is at least one x_3 for which x_3 is an office clerk if and only if x_3 will sleep, and that for all x_3, x_3 is not an office clerk we can infer that there is at least one x_3 for which x_3 will not sleep via existential biconditional elimination. Therefore, the answer is yes.