Let $X$ be a random variable with strictly increasing $F(t)$ cumulative distribution function (with inverse function $F^{-1}(t)$). We can show that $F(X)$ ~ $unif(0,1)$:
$$P(F(X)≤t)=P(X≤F^{-1}(t))=F(F^{-1}(t))=t$$ for $0≤t≤1$ (plus a few more easy steps...)
Is there any intuitive explanation why $F(X)$ ~ $unif(0,1)$?
Related question, but unanswered: All Cdfs have a uniform distribution on [0, 1]?
Since $F$ is strictly increasing, it's invertible. Say $Y$ has another invertible cdf, $G$. Then $F^{-1}(X)$ and $G^{-1}(Y)$ have the same distribution. And if $Y$ happens to be uniformly distributed, $G$ is just the identity.