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