Consider the following premises. There is at least one x_6 for which x_6 will work if and only if x_6 will take a plane. For all x_6, if x_6 writes a letter, then x_6 is playing tennis. For all x_6, x_6 will not play tennis. For all x_6, if x_6 plays tennis, then x_6 is taking a plane. For all x_6, if x_6 takes a plane, then x_6 is writing a letter. Can we infer the following from them? If we can, name the inference rule being used: For all x_6, x_6 will work.
 From the fact that for all x_6, if x_6 takes a plane, then x_6 is writing a letter, and that for all x_6, if x_6 writes a letter, then x_6 is playing tennis we can infer that for all x_6, if x_6 takes a plane, then x_6 is playing tennis via universal transitivity. From the fact that for all x_6, if x_6 plays tennis, then x_6 is taking a plane, and that for all x_6, if x_6 takes a plane, then x_6 is playing tennis we can infer that for all x_6, x_6 will play tennis if and only if x_6 will take a plane via universal biconditional introduction. From the fact that for all x_6, x_6 will play tennis if and only if x_6 will take a plane, and that for all x_6, x_6 will not play tennis we can infer that for all x_6, x_6 will not take a plane via universal biconditional elimination. Finally, from the fact that there is at least one x_6 for which x_6 will work if and only if x_6 will take a plane, and that for all x_6, x_6 will not take a plane we can infer that there is at least one x_6 for which x_6 will not work via existential biconditional elimination, which contradicts that For all x_6, x_6 will work. Therefore, the answer is no.