Consider the following premises. There is at least one x_10 for which x_10 will sleep or x_10 will not go running. For all x_10, if x_10 sleeps, then x_10 is running. For all x_10, if x_10 sleeps, then x_10 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_10 for which if x_10 goes running, then x_10 is playing squash.
 From the fact that there is at least one x_10 for which x_10 will sleep or x_10 will not go running we can infer that there is at least one x_10 for which if x_10 goes running, then x_10 is sleeping via existential rewriting. From the fact that there is at least one x_10 for which if x_10 goes running, then x_10 is sleeping, and that for all x_10, if x_10 sleeps, then x_10 is running we can infer that there is at least one x_10 for which x_10 will go running if and only if x_10 will sleep via existential biconditional introduction. From the fact that there is at least one x_10 for which x_10 will go running if and only if x_10 will sleep we can infer that there is at least one x_10 for which if x_10 goes running, then x_10 is sleeping via existential biconditional elimination. Finally, from the fact that there is at least one x_10 for which if x_10 goes running, then x_10 is sleeping, and that for all x_10, if x_10 sleeps, then x_10 is playing squash we can infer that there is at least one x_10 for which if x_10 goes running, then x_10 is playing squash via existential transitivity. Therefore, the answer is yes.