Consider the following premises. forall x_6: P_6(x_6) <-> S(x_6). forall x_6: P_6(x_6) or ~S(x_6). Can we infer exists x_6: ~(S(x_6) <-> P_6(x_6)) from them? If possible, name the inference rules being used at each step.
 From forall x_6: P_6(x_6) or ~S(x_6) we can infer forall x_6: S(x_6) -> P_6(x_6) via universal rewriting. From forall x_6: P_6(x_6) <-> S(x_6) we can infer forall x_6: P_6(x_6) -> S(x_6) via universal biconditional elimination. Finally, from forall x_6: S(x_6) -> P_6(x_6), forall x_6: P_6(x_6) -> S(x_6) we can infer forall x_6: S(x_6) <-> P_6(x_6) via universal biconditional introduction, which contradicts exists x_6: ~(S(x_6) <-> P_6(x_6)). Therefore, the answer is no.