Consider the following premises. For all x_4, x_4 will sleep if and only if x_4 will write a letter. There is at least one x_4 for which if x_4 sleeps, then x_4 is an artist. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_4 for which if x_4 writes a letter, then x_4 is an artist.
 From the fact that for all x_4, x_4 will sleep if and only if x_4 will write a letter we can infer that for all x_4, if x_4 writes a letter, then x_4 is sleeping via universal biconditional elimination. Finally, from the fact that for all x_4, if x_4 writes a letter, then x_4 is sleeping, and that there is at least one x_4 for which if x_4 sleeps, then x_4 is an artist we can infer that there is at least one x_4 for which if x_4 writes a letter, then x_4 is an artist via existential transitivity. Therefore, the answer is yes.