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