Suppose we are given two filtered probability spaces $(\Omega, \mathcal F, (\mathcal F_t)_t, \mathbb P)$ and $(\Omega', \mathcal F', (\mathcal F'_t)_t, \mathbb P')$. On $\Omega$ (respectively $\Omega'$) we are a given a standard Wiener process $W$ (respectively $W'$) adapted to the filtration. Suppose also to have a stochastic process $(X_t)_t$ on $\Omega$ such that it makes sense to define its Ito integral $\int_0^T X(t)dW(t)$. Suppose similarly to have a stochastic process $(X'_t)_t$ on $\Omega'$ such that it makes sense to define its Ito integral $\int_0^T X'(t)dW'(t)$. Finally, assume that $X_t \stackrel{d}{=}X'_t$ for any $t$.
Question: is it true that $$\int_0^T X(t)dW(t) \stackrel{d}{=} \int_0^T X'(t)dW'(t)$$ holds? Intuitively, it seems true to me, but I cannot find a proof for it.