Consider the following premises. For all x_8, if x_8 climbs a mountain, then x_8 is happy. For all x_8, if x_8 goes running, then x_8 is climbing a mountain. There is at least one x_8 for which if x_8 goes running, then x_8 is happy. For all x_8, if x_8 were happy, then x_8 is running. Can we infer the following from them? If we can, name the inference rule being used: For all x_8, if is not true that x_8 will go running if and only if x_8 is happy.
 From the fact that for all x_8, if x_8 goes running, then x_8 is climbing a mountain, and that for all x_8, if x_8 climbs a mountain, then x_8 is happy we can infer that for all x_8, if x_8 goes running, then x_8 is happy via universal transitivity. From the fact that there is at least one x_8 for which if x_8 goes running, then x_8 is happy, and that for all x_8, if x_8 were happy, then x_8 is running we can infer that there is at least one x_8 for which x_8 will go running if and only if x_8 is happy via existential biconditional introduction. From the fact that there is at least one x_8 for which x_8 will go running if and only if x_8 is happy we can infer that there is at least one x_8 for which if x_8 were happy, then x_8 is running via existential biconditional elimination. Finally, from the fact that for all x_8, if x_8 goes running, then x_8 is happy, and that there is at least one x_8 for which if x_8 were happy, then x_8 is running we can infer that there is at least one x_8 for which x_8 will go running if and only if x_8 is happy via existential biconditional introduction, which contradicts that For all x_8, if is not true that x_8 will go running if and only if x_8 is happy. Therefore, the answer is no.