Let $G$ be a causal directed acyclic graph, let $A$ be a treatment set, let $Y$ be an outcome set, and let $Z$ be a measured covariate set disjoint from $A\cup Y$. Fix an intervention value $a$, and form the single-world intervention graph $G(a)$ by splitting each treatment node into a random treatment node $A_i$ and a fixed intervention node $a_i$. Suppose that no element of $Z$ is a descendant in $G(a)$ of any fixed intervention node $a_i$ on a directed path from $a_i$ to a node in $Y(a)$, and suppose that \begin{align*} Y(a)\perp\!\!\!\perp A\mid Z \end{align*} by d-separation in $G(a)$, where $A$ denotes the collection of random treatment nodes and $Y(a)$ denotes the outcome nodes under the fixed intervention. Under consistency and positivity, the intervention law is identified by \begin{align*} \mathbb P(Y\in B\mid do(A=a)) &=\int \mathbb P(Y\in B\mid A=a,Z=z)\,d\mu_Z(z). \end{align*} This criterion refines the back-door criterion by allowing some valid adjustment sets that do not satisfy the original back-door wording.