Consider the following premises. For all x_10, if x_10 plays tennis, then x_10 is making tea. There is at least one x_10 for which if x_10 makes tea, then x_10 is a carpenter. For all x_10, if x_10 were a carpenter, then x_10 is playing tennis. For all x_10, x_10 will not make tea. For all x_10, if x_10 plays tennis, then x_10 is making tea. Can we infer the following from them? If we can, name the inference rule being used: For all x_10, x_10 will play tennis.
 From the fact that there is at least one x_10 for which if x_10 makes tea, then x_10 is a carpenter, and that for all x_10, if x_10 were a carpenter, then x_10 is playing tennis we can infer that there is at least one x_10 for which if x_10 makes tea, then x_10 is playing tennis via existential transitivity. From the fact that there is at least one x_10 for which if x_10 makes tea, then x_10 is playing tennis, and that for all x_10, if x_10 plays tennis, then x_10 is making tea we can infer that there is at least one x_10 for which x_10 will make tea if and only if x_10 will play tennis via existential biconditional introduction. From the fact that there is at least one x_10 for which x_10 will make tea if and only if x_10 will play tennis we can infer that there is at least one x_10 for which if x_10 makes tea, then x_10 is playing tennis via existential biconditional elimination. From the fact that for all x_10, if x_10 plays tennis, then x_10 is making tea, and that there is at least one x_10 for which if x_10 makes tea, then x_10 is playing tennis we can infer that there is at least one x_10 for which x_10 will play tennis if and only if x_10 will make tea via existential biconditional introduction. Finally, from the fact that there is at least one x_10 for which x_10 will play tennis if and only if x_10 will make tea, and that for all x_10, x_10 will not make tea we can infer that there is at least one x_10 for which x_10 will not play tennis via existential biconditional elimination, which contradicts that For all x_10, x_10 will play tennis. Therefore, the answer is no.