Consider the following premises. For all z_4, z_4 will write a letter if and only if z_4 is a lawyer. For all z_4, z_4 will not sleep. There is at least one z_4 for which if z_4 were a lawyer, then z_4 is sleeping. Can we infer the following from them? If we can, name the inference rule being used: For all z_4, z_4 will write a letter.
 From the fact that there is at least one z_4 for which if z_4 were a lawyer, then z_4 is sleeping, and that for all z_4, z_4 will not sleep we can infer that there is at least one z_4 for which z_4 is not a lawyer via existential modus tollens. Finally, from the fact that for all z_4, z_4 will write a letter if and only if z_4 is a lawyer, and that there is at least one z_4 for which z_4 is not a lawyer we can infer that there is at least one z_4 for which z_4 will not write a letter via existential biconditional elimination, which contradicts that For all z_4, z_4 will write a letter. Therefore, the answer is no.