Estimation in a model with infinite dimensional nuisance parameter

Let $X_1$ be a random variable with density function $f(t)$, $\Psi(t)$ be an increasing absolutely continuous function, $\Phi(t)$ be the inverse function, random variable $X_2$ be defined by $X_2=\Phi(X_1)$. We consider the maximum likelihood estimator for density $\psi$ of function $\Psi$ as we observe two independent samples from the distribution of $X_1$ and $X_2$. Under appropriate conditions on the involved distributions, we prove the consistency of maximum likelihood estimator.