Consider the following premises. For all x_7, x_7 is a bartender if and only if x_7 will play squash. If Barbara plays squash, then Mary is cooking. Joseph is poor if and only if Mary is cooking. Joseph is not poor. Can we infer the following from them? If we can, name the inference rule being used: Barbara is a bartender.
 From the fact that Joseph is poor if and only if Mary is cooking, and that Joseph is not poor we can infer that Mary is not cooking via biconditional elimination. From the fact that for all x_7, x_7 is a bartender if and only if x_7 will play squash we can infer that Barbara is a bartender if and only if Barbara is playing squash via universal instantiation. From the fact that if Barbara plays squash, then Mary is cooking, and that Mary is not cooking we can infer that Barbara is not playing squash via modus tollens. Finally, from the fact that Barbara is a bartender if and only if Barbara is playing squash, and that Barbara is not playing squash we can infer that Barbara is not a bartender via biconditional elimination, which contradicts that Barbara is a bartender. Therefore, the answer is no.