Let \(\phi(\kappa)\) be a local large cardinal axiom if and only if it is of the form \(\exists\kappa_0…\kappa_n(\phi(\kappa,\kappa_0…))\land\forall i\le n(\kappa_i\in C^{(\alpha_i)})\), such that \(V_{\kappa_i}\prec_{\alpha_{i+1}} V_{\kappa_{i+1}}\).