Consider the following premises. exists x_6: W_3(x_6) <-> R(x_6). forall x_6: Q_6(x_6) -> R_6(x_6). forall x_6: R(x_6) -> Q_6(x_6). forall x_6: R_6(x_6) -> R(x_6). forall x_6: ~R_6(x_6). Can we infer exists x_6: ~W_3(x_6) from them? If possible, name the inference rules being used at each step.
 From forall x_6: R(x_6) -> Q_6(x_6), forall x_6: Q_6(x_6) -> R_6(x_6) we can infer forall x_6: R(x_6) -> R_6(x_6) via universal transitivity. From forall x_6: R_6(x_6) -> R(x_6), forall x_6: R(x_6) -> R_6(x_6) we can infer forall x_6: R_6(x_6) <-> R(x_6) via universal biconditional introduction. From forall x_6: R_6(x_6) <-> R(x_6), forall x_6: ~R_6(x_6) we can infer forall x_6: ~R(x_6) via universal biconditional elimination. Finally, from exists x_6: W_3(x_6) <-> R(x_6), forall x_6: ~R(x_6) we can infer exists x_6: ~W_3(x_6) via existential biconditional elimination. Therefore, the answer is yes.