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