Consider the following premises. forall y: S_1(y) <-> P(y). forall y: P(y) <-> S_1(y). Can we infer exists y: ~(S_1(y) -> P(y)) from them? If possible, name the inference rules being used at each step.
 From forall y: P(y) <-> S_1(y) we can infer forall y: P(y) -> S_1(y) via universal biconditional elimination. From forall y: S_1(y) <-> P(y) we can infer forall y: S_1(y) -> P(y) via universal biconditional elimination. From forall y: P(y) -> S_1(y), forall y: S_1(y) -> P(y) we can infer forall y: P(y) <-> S_1(y) via universal biconditional introduction. Finally, from forall y: P(y) <-> S_1(y) we can infer forall y: S_1(y) -> P(y) via universal biconditional elimination, which contradicts exists y: ~(S_1(y) -> P(y)). Therefore, the answer is no.