Consider the following premises. Joseph is an electrician if and only if Karen is a scientist. Karen is a scientist or Patricia is not bored. For all x_12, x_12 is bored if and only if x_12 is a carpenter. Joseph is not an electrician. Can we infer the following from them? If we can, name the inference rule being used: Patricia is a carpenter.
 From the fact that Joseph is an electrician if and only if Karen is a scientist, and that Joseph is not an electrician we can infer that Karen is not a scientist via biconditional elimination. From the fact that for all x_12, x_12 is bored if and only if x_12 is a carpenter we can infer that Patricia is bored if and only if Patricia is a carpenter via universal instantiation. From the fact that Karen is a scientist or Patricia is not bored we can infer that if Patricia were bored, then Karen is a scientist via rewriting. From the fact that if Patricia were bored, then Karen is a scientist, and that Karen is not a scientist we can infer that Patricia is not bored via modus tollens. Finally, from the fact that Patricia is bored if and only if Patricia is a carpenter, and that Patricia is not bored we can infer that Patricia is not a carpenter via biconditional elimination, which contradicts that Patricia is a carpenter. Therefore, the answer is no.