Consider the following premises. forall x_2: P(x_2) <-> S(x_2). forall x_2: P(x_2) -> S(x_2). forall x_2: ~P(x_2). forall x_2: S(x_2) <-> Q_4(x_2). Can we infer exists x_2: Q_4(x_2) from them? If possible, name the inference rules being used at each step.
 From forall x_2: P(x_2) <-> S(x_2) we can infer forall x_2: S(x_2) -> P(x_2) via universal biconditional elimination. From forall x_2: S(x_2) -> P(x_2), forall x_2: P(x_2) -> S(x_2) we can infer forall x_2: S(x_2) <-> P(x_2) via universal biconditional introduction. From forall x_2: S(x_2) <-> P(x_2), forall x_2: ~P(x_2) we can infer forall x_2: ~S(x_2) via universal biconditional elimination. Finally, from forall x_2: S(x_2) <-> Q_4(x_2), forall x_2: ~S(x_2) we can infer forall x_2: ~Q_4(x_2) via universal biconditional elimination, which contradicts exists x_2: Q_4(x_2). Therefore, the answer is no.