Consider the following premises. exists x_1: R(x_1) -> W_3(x_1). forall x_1: W_3(x_1) -> R(x_1). forall x_1: R(x_1) or ~W_3(x_1). Can we infer forall x_1: ~(R(x_1) <-> W_3(x_1)) from them? If possible, name the inference rules being used at each step.
 From forall x_1: W_3(x_1) -> R(x_1), exists x_1: R(x_1) -> W_3(x_1) we can infer exists x_1: W_3(x_1) <-> R(x_1) via existential biconditional introduction. From forall x_1: R(x_1) or ~W_3(x_1) we can infer forall x_1: W_3(x_1) -> R(x_1) via universal rewriting. From exists x_1: W_3(x_1) <-> R(x_1) we can infer exists x_1: R(x_1) -> W_3(x_1) via existential biconditional elimination. Finally, from exists x_1: R(x_1) -> W_3(x_1), forall x_1: W_3(x_1) -> R(x_1) we can infer exists x_1: R(x_1) <-> W_3(x_1) via existential biconditional introduction, which contradicts forall x_1: ~(R(x_1) <-> W_3(x_1)). Therefore, the answer is no.