proof that p-value is uniformly distributed when the null hypothesis is true

Question

this is a follow up to the following response: https://stats.stackexchange.com/a/11288/221142

the response makes sense if $P=F(T)$ but what if the hypothesis you're testing requires $P$ to be calculated differently for example $P=1-F(T)$ ?

Answer 1

If $F(T)$ is uniformly distributed, so is $1 - F(T)$.

$$ P(1 - F(T) < t) = P(F(T) > 1 - t) = 1 - P(F(T) \leq 1 - t) = 1 - (1 - t) = t $$

Where in the second to last inequality, we use that $F(T)$ is uniformly distributed.