Consider the following premises. forall z_3: Q(z_3) -> P(z_3). forall z_3: P(z_3) <-> Q(z_3). Can we infer exists z_3: ~(P(z_3) -> Q(z_3)) from them? If possible, name the inference rules being used at each step.
 From forall z_3: P(z_3) <-> Q(z_3) we can infer forall z_3: P(z_3) -> Q(z_3) via universal biconditional elimination. From forall z_3: P(z_3) -> Q(z_3), forall z_3: Q(z_3) -> P(z_3) we can infer forall z_3: P(z_3) <-> Q(z_3) via universal biconditional introduction. Finally, from forall z_3: P(z_3) <-> Q(z_3) we can infer forall z_3: P(z_3) -> Q(z_3) via universal biconditional elimination, which contradicts exists z_3: ~(P(z_3) -> Q(z_3)). Therefore, the answer is no.