Consider the following premises. forall y_3: S_3(y_3) -> P_3(y_3). forall y_3: P_3(y_3) -> P(y_3). forall y_3: P(y_3) -> S_3(y_3). Can we infer exists y_3: ~(S_3(y_3) -> P(y_3)) from them? If possible, name the inference rules being used at each step.
 From forall y_3: S_3(y_3) -> P_3(y_3), forall y_3: P_3(y_3) -> P(y_3) we can infer forall y_3: S_3(y_3) -> P(y_3) via universal transitivity. From forall y_3: S_3(y_3) -> P(y_3), forall y_3: P(y_3) -> S_3(y_3) we can infer forall y_3: S_3(y_3) <-> P(y_3) via universal biconditional introduction. Finally, from forall y_3: S_3(y_3) <-> P(y_3) we can infer forall y_3: S_3(y_3) -> P(y_3) via universal biconditional elimination, which contradicts exists y_3: ~(S_3(y_3) -> P(y_3)). Therefore, the answer is no.