 A substitution \(\sigma\) for \(L\) is a map from the set of variables \(V\) to the set of formulas of \(L\). It tells us which formula must replace which variable when we perform the substitution. If \(p\) is a variable, then \(\sigma(p)\) denotes the formula that the substitution \(\sigma\) assigns to \(p\). The result of applying a substitution \(\sigma\) to a formula \(\phi\) is the formula \(\bsigma(\phi)\) obtained from \(\phi\) by simultaneously replacing the variables in \(\phi\), say \(p_1 , \ldots ,p_k\), by, respectively, the formulas \(\sigma(p_1), \ldots ,\sigma(p_k)\). In this way, a substitution \(\sigma\) gives a unique map \(\bsigma\) from the set of formulas to itself that satisfies