Consider the following premises. forall x_12: P_12(x_12) <-> S_4(x_12). exists x_12: P_12(x_12). forall x_12: T(x_12). Can we infer forall x_12: ~(T(x_12) and S_4(x_12)) from them? If possible, name the inference rules being used at each step.
 From forall x_12: P_12(x_12) <-> S_4(x_12) we can infer forall x_12: P_12(x_12) -> S_4(x_12) via universal biconditional elimination. From forall x_12: P_12(x_12) -> S_4(x_12), exists x_12: P_12(x_12) we can infer exists x_12: S_4(x_12) via existential modus ponens. Finally, from forall x_12: T(x_12), exists x_12: S_4(x_12) we can infer exists x_12: T(x_12) and S_4(x_12) via existential conjunction, which contradicts forall x_12: ~(T(x_12) and S_4(x_12)). Therefore, the answer is no.