Consider the following premises. exists x: P_3(x) or ~U_2(x). forall x: P_3(x) -> P(x). forall x: P(x) -> P_3(x). forall x: P(x) -> U_2(x). Can we infer forall x: ~(P_3(x) <-> U_2(x)) from them? If possible, name the inference rules being used at each step.
 From exists x: P_3(x) or ~U_2(x) we can infer exists x: U_2(x) -> P_3(x) via existential rewriting. Finally, from forall x: P_3(x) -> P(x), forall x: P(x) -> P_3(x) we can infer forall x: P_3(x) <-> P(x) via universal biconditional introduction, which contradicts forall x: ~(P_3(x) <-> U_2(x)). Therefore, the answer is no.