Consider the following premises. For all x_6, if x_6 were a nurse, then x_6 is playing squash. There is at least one x_6 for which if x_6 plays squash, then x_6 is running. For all x_6, x_6 will go running if and only if x_6 is a soldier. For all x_6, x_6 is a soldier or x_6 is a nurse. Can we infer the following from them? If we can, name the inference rule being used: For all x_6, x_6 will not go running.
 From the fact that for all x_6, if x_6 were a nurse, then x_6 is playing squash, and that there is at least one x_6 for which if x_6 plays squash, then x_6 is running we can infer that there is at least one x_6 for which if x_6 were a nurse, then x_6 is running via existential transitivity. From the fact that for all x_6, x_6 will go running if and only if x_6 is a soldier we can infer that for all x_6, if x_6 were a soldier, then x_6 is running via universal biconditional elimination. Finally, from the fact that for all x_6, if x_6 were a soldier, then x_6 is running, that there is at least one x_6 for which if x_6 were a nurse, then x_6 is running, and that for all x_6, x_6 is a soldier or x_6 is a nurse we can infer that there is at least one x_6 for which x_6 will go running via existential disjunction elimination, which contradicts that For all x_6, x_6 will not go running. Therefore, the answer is no.