Consider the following premises. forall x: P(x) -> W(x). forall x: S_3(x) or R(x). forall x: R(x) -> P(x). exists x: W(x) or ~S_3(x). Can we infer exists x: W(x) from them? If possible, name the inference rules being used at each step.
 From exists x: W(x) or ~S_3(x) we can infer exists x: S_3(x) -> W(x) via existential rewriting. From forall x: R(x) -> P(x), forall x: P(x) -> W(x) we can infer forall x: R(x) -> W(x) via universal transitivity. Finally, from exists x: S_3(x) -> W(x), forall x: R(x) -> W(x), forall x: S_3(x) or R(x) we can infer exists x: W(x) via existential disjunction elimination. Therefore, the answer is yes.