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