Definition 12: Given statements \phi and \psi and a variable x , we may refer to the universally quantified implication \forall x (\phi \Rightarrow \psi), \hspace{5mm} \textup{i.e.} \hspace{5mm} \textup{"for all } x \textup{, we have that } \phi \textup{ implies } \psi \textup{"}, as a bounded universal statement. We may also express this by saying \textup{"for all } x \textup{ such that } \phi \textup{, we have } \psi \textup{"}. Given some definition which specifies a shorthand of the form “ x is a \mathsf{D} ” for a statement \phi , we may express \forall x (\phi \Rightarrow \psi) by \textup{"for all } \mathsf{D}\textup{'s } x \textup{, we have } \psi \textup{"}.