Condition (1) in the definition of algebraizable logic can be reformulated as \[\Gamma \vdash_{\bL } \phi\txtiff \iEq(\Gamma) \vDash_{\bK } \iEq(\phi)\] and condition (2) as \[p \approx q \vDash_{\bK } \iEq(\Delta(p, q)) \textrm{ and } \iEq(\Delta(p, q)) \vDash_{\bK } p \approx q.\]