Consider the following premises. forall x_4: P_1(x_4) <-> U(x_4). exists x_4: U(x_4) <-> S(x_4). Can we infer forall x_4: ~(S(x_4) -> P_1(x_4)) from them? If possible, name the inference rules being used at each step.
 From exists x_4: U(x_4) <-> S(x_4) we can infer exists x_4: S(x_4) -> U(x_4) via existential biconditional elimination. Finally, from forall x_4: P_1(x_4) <-> U(x_4) we can infer forall x_4: U(x_4) -> P_1(x_4) via universal biconditional elimination, which contradicts forall x_4: ~(S(x_4) -> P_1(x_4)). Therefore, the answer is no.