Consider the following premises. For all x_3, if x_3 were excited, then x_3 is fast. Mary is not a sailor. For all x_3, if x_3 were fast, then x_3 is a sailor. Mary is a sailor if and only if Mary is excited. Can we infer the following from them? If we can, name the inference rule being used: Mary is not excited.
 From the fact that for all x_3, if x_3 were excited, then x_3 is fast, and that for all x_3, if x_3 were fast, then x_3 is a sailor we can infer that for all x_3, if x_3 were excited, then x_3 is a sailor via universal transitivity. From the fact that for all x_3, if x_3 were excited, then x_3 is a sailor we can infer that if Mary were excited, then Mary is a sailor via universal instantiation. From the fact that Mary is a sailor if and only if Mary is excited we can infer that if Mary were a sailor, then Mary is excited via biconditional elimination. From the fact that if Mary were excited, then Mary is a sailor, and that if Mary were a sailor, then Mary is excited we can infer that Mary is excited if and only if Mary is a sailor via biconditional introduction. Finally, from the fact that Mary is excited if and only if Mary is a sailor, and that Mary is not a sailor we can infer that Mary is not excited via biconditional elimination. Therefore, the answer is yes.