Consider the following premises. forall x: P(x) -> T_3(x). exists x: P(x) or ~U_4(x). forall x: T_3(x) -> U_4(x). Can we infer forall x: ~(P(x) <-> U_4(x)) from them? If possible, name the inference rules being used at each step.
 From forall x: P(x) -> T_3(x), forall x: T_3(x) -> U_4(x) we can infer forall x: P(x) -> U_4(x) via universal transitivity. From exists x: P(x) or ~U_4(x) we can infer exists x: U_4(x) -> P(x) via existential rewriting. Finally, from forall x: P(x) -> U_4(x), exists x: U_4(x) -> P(x) we can infer exists x: P(x) <-> U_4(x) via existential biconditional introduction, which contradicts forall x: ~(P(x) <-> U_4(x)). Therefore, the answer is no.