Let \((a(\cdot, \cdot), D)\) be a regular Dirichlet form of diffusion type on \(L^2(X, m)\) satisfying the following hypotheses: a) the metric topology induced by the intrinsic distance on \(X\) is equivalent to the original topology on \(X\); b) the measure \(m\) is doubling with respect to the intrinsic balls; c) \(X_0\) is a relatively compact open subset of \(X\) and \(a(\cdot, \cdot)\) satisfies a Poincaré inequality on \(X_0\). The author proves that, for any \(O \subset X_0\), any local solution \(u\) of the problem \[ a(u, v)=0, \qquad \forall v\in D_0(O) \] satisfies a Meyer estimate.