Consider the following premises. forall z_4: Q(z_4) or ~P(z_4). forall z_4: Q(z_4) -> P(z_4). forall z_4: P(z_4) -> Q(z_4). Can we infer exists z_4: ~(P(z_4) <-> Q(z_4)) from them? If possible, name the inference rules being used at each step.
 From forall z_4: Q(z_4) -> P(z_4), forall z_4: P(z_4) -> Q(z_4) we can infer forall z_4: Q(z_4) <-> P(z_4) via universal biconditional introduction. From forall z_4: Q(z_4) <-> P(z_4) we can infer forall z_4: Q(z_4) -> P(z_4) via universal biconditional elimination. From forall z_4: Q(z_4) or ~P(z_4) we can infer forall z_4: P(z_4) -> Q(z_4) via universal rewriting. Finally, from forall z_4: P(z_4) -> Q(z_4), forall z_4: Q(z_4) -> P(z_4) we can infer forall z_4: P(z_4) <-> Q(z_4) via universal biconditional introduction, which contradicts exists z_4: ~(P(z_4) <-> Q(z_4)). Therefore, the answer is no.