 \(\Gamma \vdash_{\bILL } \phi \txtiff\) for every \(\bA \in \bILsubZ\) and every valuation \(v\) on \(\bA\), if \(1^{\bA } \le^{\bA } \bv(\psi)\) for all \(\psi \in \Gamma \), then \(1^{\bA } \le^{\bA } \bv(\phi)\).