Consider the following premises. If Joseph were a janitor, then Joseph is writing a letter. For all x_17, if x_17 were a taxi driver, then x_17 is cooking. Joseph is writing a letter or Jennifer is a taxi driver. If Joseph writes a letter, then Joseph is a janitor. If Jennifer cooks, then Joseph is a janitor. Can we infer the following from them? If we can, name the inference rule being used: Joseph is a janitor.
 From the fact that for all x_17, if x_17 were a taxi driver, then x_17 is cooking we can infer that if Jennifer were a taxi driver, then Jennifer is cooking via universal instantiation. From the fact that if Jennifer were a taxi driver, then Jennifer is cooking, and that if Jennifer cooks, then Joseph is a janitor we can infer that if Jennifer were a taxi driver, then Joseph is a janitor via transitivity. From the fact that if Joseph writes a letter, then Joseph is a janitor, and that if Joseph were a janitor, then Joseph is writing a letter we can infer that Joseph is writing a letter if and only if Joseph is a janitor via biconditional introduction. From the fact that Joseph is writing a letter if and only if Joseph is a janitor we can infer that if Joseph writes a letter, then Joseph is a janitor via biconditional elimination. Finally, from the fact that if Joseph writes a letter, then Joseph is a janitor, that if Jennifer were a taxi driver, then Joseph is a janitor, and that Joseph is writing a letter or Jennifer is a taxi driver we can infer that Joseph is a janitor via disjunction elimination. Therefore, the answer is yes.