Consider the following premises. forall x: P(x) or ~S(x). forall x: P(x) -> S(x). exists x: ~P(x). Can we infer exists x: ~S(x) from them? If possible, name the inference rules being used at each step.
 From forall x: P(x) or ~S(x) we can infer forall x: S(x) -> P(x) via universal rewriting. From forall x: S(x) -> P(x), forall x: P(x) -> S(x) we can infer forall x: S(x) <-> P(x) via universal biconditional introduction. Finally, from forall x: S(x) <-> P(x), exists x: ~P(x) we can infer exists x: ~S(x) via existential biconditional elimination. Therefore, the answer is yes.