Consider the following premises. For all x_12, if x_12 writes a letter, then x_12 is a poet. For all x_12, if x_12 were a poet, then x_12 is taking a plane. For all x_12, if x_12 were a poet, then x_12 is cooking. For all x_12, x_12 will not cook. There is at least one x_12 for which if x_12 takes a plane, then x_12 is writing a letter. For all x_12, if x_12 cooks, then x_12 is a poet. Can we infer the following from them? If we can, name the inference rule being used: For all x_12, x_12 will write a letter.
 From the fact that for all x_12, if x_12 cooks, then x_12 is a poet, and that for all x_12, if x_12 were a poet, then x_12 is cooking we can infer that for all x_12, x_12 will cook if and only if x_12 is a poet via universal biconditional introduction. From the fact that for all x_12, if x_12 were a poet, then x_12 is taking a plane, and that there is at least one x_12 for which if x_12 takes a plane, then x_12 is writing a letter we can infer that there is at least one x_12 for which if x_12 were a poet, then x_12 is writing a letter via existential transitivity. From the fact that for all x_12, x_12 will cook if and only if x_12 is a poet, and that for all x_12, x_12 will not cook we can infer that for all x_12, x_12 is not a poet via universal biconditional elimination. From the fact that there is at least one x_12 for which if x_12 were a poet, then x_12 is writing a letter, and that for all x_12, if x_12 writes a letter, then x_12 is a poet we can infer that there is at least one x_12 for which x_12 is a poet if and only if x_12 will write a letter via existential biconditional introduction. Finally, from the fact that there is at least one x_12 for which x_12 is a poet if and only if x_12 will write a letter, and that for all x_12, x_12 is not a poet we can infer that there is at least one x_12 for which x_12 will not write a letter via existential biconditional elimination, which contradicts that For all x_12, x_12 will write a letter. Therefore, the answer is no.