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