I am trying to solve the following problem from a book on mathematical statistics:
Suppose $\mathcal{P}=\{f(x|\theta_1,\theta_2)|\theta=(\theta_1,\theta_2)\in\Theta_1\times\Theta_2\}$ is a family of non-vanishing pdfs on $\mathbb{R}^d$, meaning that $f(x|\theta)\neq0$ for any $x\in\mathbb{R}^d,\theta\in\Theta_1\times\Theta_2$.
Assume the statistic $T=(T_1,T_2)$ satisfies the following properties:
$\cdot$ Assume $\theta_1\in\Theta_1$ is known, then $T_2$ is sufficient for $\theta_2$;
$\cdot$ Assume $\theta_2\in\Theta_2$ is known, then $T_1$ is sufficient for $\theta_1$.
Show that $T$ is sufficient for $\theta$.
By using factorization theorem, I derived $$f(x|\theta)=p_1(T_1(x),\theta_1,\theta_2)h_1(x,\theta_2)=p_2(T_2(x),\theta_1,\theta_2)h_2(x,\theta_1).$$ However, I don't know how to proceed. The solution says in the equality above, $p_1$ has nothing to do with $\theta_2$, so we may write $p_1(T_1(x),\theta_1,\theta_2)=p_1(T_1(x),\theta_1)$. I do know if this holds, the conclusion is true. But why is this fact true? If it is not true, what extra conditions do I need to prove the desired conclusion? Or maybe the statement itself is wrong?