Consider the following premises. It is overcast or Barbara is an artist. If it rains, then Patricia is a cashier. If Barbara were an artist, then Patricia is a cashier. If it is overcast, then it is raining. Can we infer the following from them? If we can, name the inference rule being used: Patricia is not a cashier.
 From the fact that if it is overcast, then it is raining, and that if it rains, then Patricia is a cashier we can infer that if it is overcast, then Patricia is a cashier via transitivity. Finally, from the fact that if it is overcast, then Patricia is a cashier, that if Barbara were an artist, then Patricia is a cashier, and that it is overcast or Barbara is an artist we can infer that Patricia is a cashier via disjunction elimination, which contradicts that Patricia is not a cashier. Therefore, the answer is no.