Consider the following premises. forall y_1: T_1(y_1) <-> Q_2(y_1). forall y_1: Q_2(y_1) -> Q_3(y_1). forall y_1: Q_3(y_1) -> T_1(y_1). Can we infer exists y_1: ~(Q_3(y_1) <-> Q_2(y_1)) from them? If possible, name the inference rules being used at each step.
 From forall y_1: T_1(y_1) <-> Q_2(y_1) we can infer forall y_1: T_1(y_1) -> Q_2(y_1) via universal biconditional elimination. From forall y_1: Q_3(y_1) -> T_1(y_1), forall y_1: T_1(y_1) -> Q_2(y_1) we can infer forall y_1: Q_3(y_1) -> Q_2(y_1) via universal transitivity. Finally, from forall y_1: Q_3(y_1) -> Q_2(y_1), forall y_1: Q_2(y_1) -> Q_3(y_1) we can infer forall y_1: Q_3(y_1) <-> Q_2(y_1) via universal biconditional introduction, which contradicts exists y_1: ~(Q_3(y_1) <-> Q_2(y_1)). Therefore, the answer is no.