Consider the following premises. For all x_12, if x_12 were a bartender, then x_12 is sad. There is at least one x_12 for which x_12 is a doctor if and only if x_12 is a poet. For all x_12, x_12 is a poet or x_12 is not sad. 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 a bartender, then x_12 is a doctor.
 From the fact that for all x_12, x_12 is a poet or x_12 is not sad we can infer that for all x_12, if x_12 were sad, then x_12 is a poet via universal rewriting. From the fact that there is at least one x_12 for which x_12 is a doctor if and only if x_12 is a poet we can infer that there is at least one x_12 for which if x_12 were a poet, then x_12 is a doctor via existential biconditional elimination. From the fact that for all x_12, if x_12 were sad, then x_12 is a poet, and that there is at least one x_12 for which if x_12 were a poet, then x_12 is a doctor we can infer that there is at least one x_12 for which if x_12 were sad, then x_12 is a doctor via existential transitivity. Finally, from the fact that for all x_12, if x_12 were a bartender, then x_12 is sad, and that there is at least one x_12 for which if x_12 were sad, then x_12 is a doctor we can infer that there is at least one x_12 for which if x_12 were a bartender, then x_12 is a doctor via existential transitivity. Therefore, the answer is yes.