Consider the following premises. There is at least one x_1 for which if x_1 were happy, then x_1 is sleeping. For all x_1, x_1 is not happy. For all x_1, if x_1 sleeps, then x_1 is happy. 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 x_1 will not sleep.
 From the fact that there is at least one x_1 for which if x_1 were happy, then x_1 is sleeping, and that for all x_1, if x_1 sleeps, then x_1 is happy we can infer that there is at least one x_1 for which x_1 is happy if and only if x_1 will sleep via existential biconditional introduction. Finally, from the fact that there is at least one x_1 for which x_1 is happy if and only if x_1 will sleep, and that for all x_1, x_1 is not happy we can infer that there is at least one x_1 for which x_1 will not sleep via existential biconditional elimination. Therefore, the answer is yes.