Consider the following premises. It is overcast if and only if it is cold. It is cold or Elisabeth is not a cashier. If it is overcast, then Mary is working. Can we infer the following from them? If we can, name the inference rule being used: The fact that Elisabeth were a cashier does not imply that Mary is working.
 From the fact that it is overcast if and only if it is cold we can infer that if it is cold, then it is overcast via biconditional elimination. From the fact that it is cold or Elisabeth is not a cashier we can infer that if Elisabeth were a cashier, then it is cold via rewriting. From the fact that if it is cold, then it is overcast, and that if it is overcast, then Mary is working we can infer that if it is cold, then Mary is working via transitivity. Finally, from the fact that if Elisabeth were a cashier, then it is cold, and that if it is cold, then Mary is working we can infer that if Elisabeth were a cashier, then Mary is working via transitivity, which contradicts that The fact that Elisabeth were a cashier does not imply that Mary is working. Therefore, the answer is no.