 Suppose that the last rule applied to get \(\Gamma_1 \vdash \phi\) is \((\forall\)I). So \(\phi\) is a sentence of the form \(\forall v\theta\), and we have \(\Gamma_1 \vdash \theta (v|t)\) and \(t\) does occur in any member of \(\Gamma_1\) or in \(\theta\). The problem is that \(t\) may occur in a member of \(\Gamma_2\), and so we cannot just invoke the induction hypothesis and apply \((\forall\)I) to the result. So, let \(t'\) be a term not occurring in any sentence in \(\Gamma_2\). Let \(\Gamma'\) be the result of substituting \(t'\) for all \(t\) in \(\Gamma_2\). Then, since \(t\) does not occur in \(\Gamma_1\), \(\Gamma_1\subseteq\Gamma'\). So, the induction hypothesis gives us \(\Gamma'\vdash\theta (v|t)\), and we know that \(\Gamma'\) does not contain \(t\), so we can apply (\(\forall I\)) to get \(\Gamma'\vdash\forall v\theta\). But \(\forall v\theta\) does not contain \(t\) or \(t'\), so \(\Gamma_2\vdash\forall v\theta\) by Lemma 7.