Consider the following premises. Michael is drinking water or Joseph is a poet. John is rich if and only if Michael is drinking water. John is rich if and only if Joseph is a poet. Can we infer the following from them? If we can, name the inference rule being used: John is not rich.
 From the fact that John is rich if and only if Joseph is a poet we can infer that if Joseph were a poet, then John is rich via biconditional elimination. From the fact that John is rich if and only if Michael is drinking water we can infer that if Michael drinks water, then John is rich via biconditional elimination. Finally, from the fact that if Michael drinks water, then John is rich, that if Joseph were a poet, then John is rich, and that Michael is drinking water or Joseph is a poet we can infer that John is rich via disjunction elimination, which contradicts that John is not rich. Therefore, the answer is no.