 Theorem 8 allows us to add on premises at will. It follows that \(\Gamma \vdash \phi\) if and only if there is a subset \(\Gamma'\subseteq \Gamma\) such that \(\Gamma'\vdash \phi\). Some systems of relevant logic do not have weakening, nor does substructural logic (See the entries on  relevance logic,  substructural logics, and  linear logic).