Consider the following premises. For all x_13, if x_13 works, then x_13 is climbing a mountain. There is at least one x_13 for which if x_13 were a writer, then x_13 is working. For all x_13, if x_13 climbs a mountain, then x_13 is drinking water. For all x_13, if x_13 drinks water, then x_13 is a writer. For all x_13, x_13 will not drink water. Can we infer the following from them? If we can, name the inference rule being used: For all x_13, x_13 will work.
 From the fact that for all x_13, if x_13 works, then x_13 is climbing a mountain, and that for all x_13, if x_13 climbs a mountain, then x_13 is drinking water we can infer that for all x_13, if x_13 works, then x_13 is drinking water via universal transitivity. From the fact that for all x_13, if x_13 drinks water, then x_13 is a writer, and that there is at least one x_13 for which if x_13 were a writer, then x_13 is working we can infer that there is at least one x_13 for which if x_13 drinks water, then x_13 is working via existential transitivity. From the fact that for all x_13, if x_13 works, then x_13 is drinking water, and that there is at least one x_13 for which if x_13 drinks water, then x_13 is working we can infer that there is at least one x_13 for which x_13 will work if and only if x_13 will drink water via existential biconditional introduction. From the fact that there is at least one x_13 for which x_13 will work if and only if x_13 will drink water we can infer that there is at least one x_13 for which if x_13 works, then x_13 is drinking water via existential biconditional elimination. Finally, from the fact that there is at least one x_13 for which if x_13 works, then x_13 is drinking water, and that for all x_13, x_13 will not drink water we can infer that there is at least one x_13 for which x_13 will not work via existential modus tollens, which contradicts that For all x_13, x_13 will work. Therefore, the answer is no.