Given a logic system \(\bL\), the consequence operation \(C_{\bL}\) is substitution-invariant, which means that for every set of \(L\)-formulas \(\Gamma\) and every substitution \(\sigma, \bsigma[C_{\bL}(\Gamma)] \subseteq C_{\bL}(\bsigma[\Gamma]\)). Moreover, for every theory \(T\) of \(\bL\) we have a new consequence \(C_{\bL }^T\) operation defined as follows: \[C_{\bL }^T (\Gamma) = C_{\bL }(T \cup \Gamma)\] that is, \(C_{\bL }^T (\Gamma)\) is the set of formulas that follow from \(\Gamma\) and \(T\) according to \(\bL\). It turns out that \(T\) is closed under substitutions if and only if \(C_{\bL }^T\) is substitution-invariant.