Consider the following premises. If Jennifer were a soldier, then William is a mechanic. William is a mechanic or Mary is not a cashier. Sarah is a poet or Mary is a cashier. If Sarah were a poet, then Jennifer is a soldier. Can we infer the following from them? If we can, name the inference rule being used: William is a mechanic.
 From the fact that William is a mechanic or Mary is not a cashier we can infer that if Mary were a cashier, then William is a mechanic via rewriting. From the fact that if Sarah were a poet, then Jennifer is a soldier, and that if Jennifer were a soldier, then William is a mechanic we can infer that if Sarah were a poet, then William is a mechanic via transitivity. Finally, from the fact that if Sarah were a poet, then William is a mechanic, that if Mary were a cashier, then William is a mechanic, and that Sarah is a poet or Mary is a cashier we can infer that William is a mechanic via disjunction elimination. Therefore, the answer is yes.