Consider the following premises. For all y_4, if y_4 writes a letter, then y_4 is working. For all y_4, if y_4 works, then y_4 is a lawyer. There is at least one y_4 for which y_4 will write a letter or y_4 is not a lawyer. Can we infer the following from them? If we can, name the inference rule being used: For all y_4, if is not true that y_4 is a lawyer if and only if y_4 will write a letter.
 From the fact that for all y_4, if y_4 writes a letter, then y_4 is working, and that for all y_4, if y_4 works, then y_4 is a lawyer we can infer that for all y_4, if y_4 writes a letter, then y_4 is a lawyer via universal transitivity. From the fact that there is at least one y_4 for which y_4 will write a letter or y_4 is not a lawyer we can infer that there is at least one y_4 for which if y_4 were a lawyer, then y_4 is writing a letter via existential rewriting. Finally, from the fact that there is at least one y_4 for which if y_4 were a lawyer, then y_4 is writing a letter, and that for all y_4, if y_4 writes a letter, then y_4 is a lawyer we can infer that there is at least one y_4 for which y_4 is a lawyer if and only if y_4 will write a letter via existential biconditional introduction, which contradicts that For all y_4, if is not true that y_4 is a lawyer if and only if y_4 will write a letter. Therefore, the answer is no.