Consider the following premises. forall z_4: P(z_4) <-> S_1(z_4). forall z_4: S_1(z_4) -> W(z_4). forall z_4: ~W(z_4). Can we infer forall z_4: ~P(z_4) from them? If possible, name the inference rules being used at each step.
 From forall z_4: P(z_4) <-> S_1(z_4) we can infer forall z_4: P(z_4) -> S_1(z_4) via universal biconditional elimination. From forall z_4: P(z_4) -> S_1(z_4), forall z_4: S_1(z_4) -> W(z_4) we can infer forall z_4: P(z_4) -> W(z_4) via universal transitivity. Finally, from forall z_4: P(z_4) -> W(z_4), forall z_4: ~W(z_4) we can infer forall z_4: ~P(z_4) via universal modus tollens. Therefore, the answer is yes.