Consider the following premises. forall z_1: U_4(z_1) -> P(z_1). forall z_1: ~W(z_1). forall z_1: S_4(z_1) -> U_4(z_1). forall z_1: W(z_1) <-> P(z_1). forall z_1: P(z_1) -> S_4(z_1). Can we infer forall z_1: ~S_4(z_1) from them? If possible, name the inference rules being used at each step.
 From forall z_1: S_4(z_1) -> U_4(z_1), forall z_1: U_4(z_1) -> P(z_1) we can infer forall z_1: S_4(z_1) -> P(z_1) via universal transitivity. From forall z_1: P(z_1) -> S_4(z_1), forall z_1: S_4(z_1) -> P(z_1) we can infer forall z_1: P(z_1) <-> S_4(z_1) via universal biconditional introduction. From forall z_1: W(z_1) <-> P(z_1), forall z_1: ~W(z_1) we can infer forall z_1: ~P(z_1) via universal biconditional elimination. Finally, from forall z_1: P(z_1) <-> S_4(z_1), forall z_1: ~P(z_1) we can infer forall z_1: ~S_4(z_1) via universal biconditional elimination. Therefore, the answer is yes.