Consider the following premises. For all x_1, if x_1 writes a letter, then x_1 is taking a plane. There is at least one x_1 for which if x_1 takes a plane, then x_1 is writing a letter. For all x_1, x_1 will not write a letter. Can we infer the following from them? If we can, name the inference rule being used: For all x_1, x_1 will take a plane.
 From the fact that there is at least one x_1 for which if x_1 takes a plane, then x_1 is writing a letter, and that for all x_1, if x_1 writes a letter, then x_1 is taking a plane we can infer that there is at least one x_1 for which x_1 will take a plane if and only if x_1 will write a letter via existential biconditional introduction. From the fact that there is at least one x_1 for which x_1 will take a plane if and only if x_1 will write a letter we can infer that there is at least one x_1 for which if x_1 takes a plane, then x_1 is writing a letter via existential biconditional elimination. Finally, from the fact that there is at least one x_1 for which if x_1 takes a plane, then x_1 is writing a letter, and that for all x_1, x_1 will not write a letter we can infer that there is at least one x_1 for which x_1 will not take a plane via existential modus tollens, which contradicts that For all x_1, x_1 will take a plane. Therefore, the answer is no.