Consider the following premises. exists x: R_3(x) or ~S(x). forall x: S(x) or ~R_3(x). Can we infer forall x: ~(S(x) <-> R_3(x)) from them? If possible, name the inference rules being used at each step.
 From forall x: S(x) or ~R_3(x) we can infer forall x: R_3(x) -> S(x) via universal rewriting. From exists x: R_3(x) or ~S(x) we can infer exists x: S(x) -> R_3(x) via existential rewriting. Finally, from exists x: S(x) -> R_3(x), forall x: R_3(x) -> S(x) we can infer exists x: S(x) <-> R_3(x) via existential biconditional introduction, which contradicts forall x: ~(S(x) <-> R_3(x)). Therefore, the answer is no.