Neyman-Pearson's lemma says that when comparing two singleton hypotheses $\{\theta_0\}$ and $\{\theta_1\}$, the likelihood ratio test of threshold $\alpha$ is uniformly more powerful than any other test of threshold $\alpha$. A proposition in my class notes states that for more complicated hypotheses we have the following:
$\Phi$ is a uniformly most powerful test of threshold $\alpha$ for the hypotheses $H_0=\{\theta_0\}$ and $H_1$ if and only if for all $\theta\in H_1$, $\Phi$ is uniformly most powerful of threshold $\alpha$ for comparing $H_0$ and $\{\theta\}$.
This seems trivial to me, but what bothers me is that it seems like it would be trivially true for a more general hypothesis $H_0$, too:
$\Phi$ is a uniformly most powerful test of threshold $\alpha$ for the hypotheses $H_0$ and $H_1$ if and only if for all $\theta_0\in H_0,\theta_1\in H_1$, $\Phi$ is uniformly most powerful of threshold $\alpha$ for comparing $\{\theta_0\}$ and $\{\theta_1\}$.
Is this not true?