Consider the following premises. Joseph is not writing a letter. For all x_10, x_10 will write a letter if and only if x_10 will play tennis. For all x_10, x_10 will write a letter if and only if x_10 will play tennis. Can we infer the following from them? If we can, name the inference rule being used: Joseph is playing tennis.
 From the fact that for all x_10, x_10 will write a letter if and only if x_10 will play tennis we can infer that for all x_10, if x_10 writes a letter, then x_10 is playing tennis via universal biconditional elimination. From the fact that for all x_10, x_10 will write a letter if and only if x_10 will play tennis we can infer that for all x_10, if x_10 plays tennis, then x_10 is writing a letter via universal biconditional elimination. From the fact that for all x_10, if x_10 writes a letter, then x_10 is playing tennis, and that for all x_10, if x_10 plays tennis, then x_10 is writing a letter we can infer that for all x_10, x_10 will write a letter if and only if x_10 will play tennis via universal biconditional introduction. From the fact that for all x_10, x_10 will write a letter if and only if x_10 will play tennis we can infer that Joseph is writing a letter if and only if Joseph is playing tennis via universal instantiation. Finally, from the fact that Joseph is writing a letter if and only if Joseph is playing tennis, and that Joseph is not writing a letter we can infer that Joseph is not playing tennis via biconditional elimination, which contradicts that Joseph is playing tennis. Therefore, the answer is no.