 \(\exists x\phi\) is true in a model \(M\) relative to a variable assignment \(\alpha\) iff (if and only if) there is some variable assignment \(\beta\) such that \(\beta\) differs from \(\alpha\) at most with respect to the value it assigns to \(x\) and such that \(\phi\) is true in \(M\) relative to assignment \(\beta\). 