Consider the following premises. For all z_4, if z_4 writes a letter, then z_4 is an electrician. For all z_4, if z_4 were an electrician, then z_4 is writing a letter. Joseph is writing a letter if and only if Patricia is playing squash. For all z_4, z_4 is not an electrician. Can we infer the following from them? If we can, name the inference rule being used: Patricia is playing squash.
 From the fact that for all z_4, if z_4 were an electrician, then z_4 is writing a letter, and that for all z_4, if z_4 writes a letter, then z_4 is an electrician we can infer that for all z_4, z_4 is an electrician if and only if z_4 will write a letter via universal biconditional introduction. From the fact that for all z_4, z_4 is an electrician if and only if z_4 will write a letter, and that for all z_4, z_4 is not an electrician we can infer that for all z_4, z_4 will not write a letter via universal biconditional elimination. From the fact that for all z_4, z_4 will not write a letter we can infer that Joseph is not writing a letter via universal instantiation. Finally, from the fact that Joseph is writing a letter if and only if Patricia is playing squash, and that Joseph is not writing a letter we can infer that Patricia is not playing squash via biconditional elimination, which contradicts that Patricia is playing squash. Therefore, the answer is no.