Consider the following premises. For all x_4, x_4 will not play a game. For all x_4, x_4 will make tea if and only if x_4 will play a game. There is at least one x_4 for which x_4 will make tea if and only if x_4 will write a letter. Can we infer the following from them? If we can, name the inference rule being used: For all x_4, x_4 will write a letter.
 From the fact that for all x_4, x_4 will make tea if and only if x_4 will play a game, and that for all x_4, x_4 will not play a game we can infer that for all x_4, x_4 will not make tea via universal biconditional elimination. Finally, from the fact that there is at least one x_4 for which x_4 will make tea if and only if x_4 will write a letter, and that for all x_4, x_4 will not make tea we can infer that there is at least one x_4 for which x_4 will not write a letter via existential biconditional elimination, which contradicts that For all x_4, x_4 will write a letter. Therefore, the answer is no.