Consider the following premises. For all x_13, x_13 will write a letter if and only if x_13 will play a game. For all x_13, if x_13 plays a game, then x_13 is writing a letter. For all x_13, x_13 will not play a game. There is at least one x_13 for which if x_13 were a carpenter, then x_13 is writing a letter. For all x_13, if x_13 goes running, then x_13 is a carpenter. Can we infer the following from them? If we can, name the inference rule being used: For all x_13, x_13 will go running.
 From the fact that for all x_13, x_13 will write a letter if and only if x_13 will play a game we can infer that for all x_13, if x_13 writes a letter, then x_13 is playing a game via universal biconditional elimination. From the fact that for all x_13, if x_13 goes running, then x_13 is a carpenter, and that there is at least one x_13 for which if x_13 were a carpenter, then x_13 is writing a letter we can infer that there is at least one x_13 for which if x_13 goes running, then x_13 is writing a letter via existential transitivity. From the fact that for all x_13, if x_13 plays a game, then x_13 is writing a letter, and that for all x_13, if x_13 writes a letter, then x_13 is playing a game we can infer that for all x_13, x_13 will play a game if and only if x_13 will write a letter via universal biconditional introduction. From the fact that for all x_13, x_13 will play a game if and only if x_13 will write a letter, and that for all x_13, x_13 will not play a game we can infer that for all x_13, x_13 will not write a letter via universal biconditional elimination. Finally, from the fact that there is at least one x_13 for which if x_13 goes running, then x_13 is writing a letter, and that for all x_13, x_13 will not write a letter we can infer that there is at least one x_13 for which x_13 will not go running via existential modus tollens, which contradicts that For all x_13, x_13 will go running. Therefore, the answer is no.