 The proof of the completeness theorem then proceeds as follows.  (4)  and  (5)  imply that for every formula \(\psi , \Gamma \vdash_{\bL } \psi\) if and only if \([\psi]\) satisfies the equation \(\delta(p) \approx \varepsilon(p)\) in the algebra \(\bFm/\theta(T)\). Thus, considering the valuation \(id\) mapping every variable \(p\) to its equivalence class \([p]\), whose extension \(\boldsymbol{id}\) to the set of all formulas is such that \(\boldsymbol{id}(\phi) = [\phi]\) for every formula \(\phi\), we have for every formula \(\psi\),