Consider the following premises. Mary is sad or Mary is not a politician. For all x_17, if x_17 were sad, then x_17 is a politician. John is not an astronaut. John is an astronaut if and only if Mary is a politician. Can we infer the following from them? If we can, name the inference rule being used: Mary is not sad.
 From the fact that for all x_17, if x_17 were sad, then x_17 is a politician we can infer that if Mary were sad, then Mary is a politician via universal instantiation. From the fact that Mary is sad or Mary is not a politician we can infer that if Mary were a politician, then Mary is sad via rewriting. From the fact that John is an astronaut if and only if Mary is a politician, and that John is not an astronaut we can infer that Mary is not a politician via biconditional elimination. From the fact that if Mary were a politician, then Mary is sad, and that if Mary were sad, then Mary is a politician we can infer that Mary is a politician if and only if Mary is sad via biconditional introduction. Finally, from the fact that Mary is a politician if and only if Mary is sad, and that Mary is not a politician we can infer that Mary is not sad via biconditional elimination. Therefore, the answer is yes.