Is $Y_t = f(X_t, \beta_t)$ $\rho$-mixing if $X_t$ and $\beta_t$ are independent and each of them are $\rho$-mixing?

Question

There is a post which says $Y_t = f(X_t, \beta_t)$ is strong mixing if $X_t$ and $\beta_t$ are independent and both strong mixing. Does this statement have any reference (book or paper)? In addition does the same apply to $\rho$-mixing sequences i.e., is $Y_t = f(X_t, \beta_t)$ $\rho$-mixing if $X_t$ and $\beta_t$ are independent and each of them are $\rho$-mixing?