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    $\begingroup$ In order to prove this you need to assume the distribution of the test statistic is the same for every element of the null hypothesis. When that's the case, this result is essentially a tautology, because the p-value is constructed to have a uniform distribution. $\endgroup$ Commented Sep 11, 2020 at 22:58
  • $\begingroup$ @whuber I almost understand what you mean about being the same, but not quite. Could you please elaborate on that? $\endgroup$ Commented Sep 11, 2020 at 23:00