Consider the following premises. forall y: T(y) or ~P(y). forall y: P_2(y) -> P(y). forall y: P(y) <-> P_2(y). Can we infer exists y: ~(P_2(y) -> T(y)) from them? If possible, name the inference rules being used at each step.
 From forall y: P(y) <-> P_2(y) we can infer forall y: P(y) -> P_2(y) via universal biconditional elimination. From forall y: T(y) or ~P(y) we can infer forall y: P(y) -> T(y) via universal rewriting. From forall y: P_2(y) -> P(y), forall y: P(y) -> P_2(y) we can infer forall y: P_2(y) <-> P(y) via universal biconditional introduction. From forall y: P_2(y) <-> P(y) we can infer forall y: P_2(y) -> P(y) via universal biconditional elimination. Finally, from forall y: P_2(y) -> P(y), forall y: P(y) -> T(y) we can infer forall y: P_2(y) -> T(y) via universal transitivity, which contradicts exists y: ~(P_2(y) -> T(y)). Therefore, the answer is no.