Consider the following premises. exists x_1: ~U_4(x_1). forall x_1: U(x_1) <-> U_4(x_1). forall x_1: U(x_1) <-> S_3(x_1). Can we infer forall x_1: S_3(x_1) from them? If possible, name the inference rules being used at each step.
 From forall x_1: U(x_1) <-> U_4(x_1), exists x_1: ~U_4(x_1) we can infer exists x_1: ~U(x_1) via existential biconditional elimination. From forall x_1: U(x_1) <-> S_3(x_1) we can infer forall x_1: S_3(x_1) -> U(x_1) via universal biconditional elimination. Finally, from forall x_1: S_3(x_1) -> U(x_1), exists x_1: ~U(x_1) we can infer exists x_1: ~S_3(x_1) via existential modus tollens, which contradicts forall x_1: S_3(x_1). Therefore, the answer is no.