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