Consider the following premises. For all x_12, x_12 is sad if and only if x_12 will write a letter. There is at least one x_12 for which if x_12 were an office clerk, then x_12 is making tea. For all x_12, if x_12 writes a letter, then x_12 is an office clerk. For all x_12, if x_12 makes tea, then x_12 is excited. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_12 for which if x_12 were sad, then x_12 is excited.
 From the fact that for all x_12, x_12 is sad if and only if x_12 will write a letter we can infer that for all x_12, if x_12 were sad, then x_12 is writing a letter via universal biconditional elimination. From the fact that for all x_12, if x_12 were sad, then x_12 is writing a letter, and that for all x_12, if x_12 writes a letter, then x_12 is an office clerk we can infer that for all x_12, if x_12 were sad, then x_12 is an office clerk via universal transitivity. From the fact that there is at least one x_12 for which if x_12 were an office clerk, then x_12 is making tea, and that for all x_12, if x_12 makes tea, then x_12 is excited we can infer that there is at least one x_12 for which if x_12 were an office clerk, then x_12 is excited via existential transitivity. Finally, from the fact that for all x_12, if x_12 were sad, then x_12 is an office clerk, and that there is at least one x_12 for which if x_12 were an office clerk, then x_12 is excited we can infer that there is at least one x_12 for which if x_12 were sad, then x_12 is excited via existential transitivity. Therefore, the answer is yes.