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