Consider the following premises. forall z_1: S(z_1) <-> S_4(z_1). forall z_1: W(z_1) -> S_4(z_1). exists z_1: W(z_1) <-> S(z_1). Can we infer exists z_1: W(z_1) -> S(z_1) from them? If possible, name the inference rules being used at each step.
 From forall z_1: S(z_1) <-> S_4(z_1) we can infer forall z_1: S_4(z_1) -> S(z_1) via universal biconditional elimination. From forall z_1: W(z_1) -> S_4(z_1), forall z_1: S_4(z_1) -> S(z_1) we can infer forall z_1: W(z_1) -> S(z_1) via universal transitivity. From exists z_1: W(z_1) <-> S(z_1) we can infer exists z_1: S(z_1) -> W(z_1) via existential biconditional elimination. From exists z_1: S(z_1) -> W(z_1), forall z_1: W(z_1) -> S(z_1) we can infer exists z_1: S(z_1) <-> W(z_1) via existential biconditional introduction. Finally, from exists z_1: S(z_1) <-> W(z_1) we can infer exists z_1: W(z_1) -> S(z_1) via existential biconditional elimination. Therefore, the answer is yes.