Homotopy equivalence and universal $G$-bundle (Showing total space is contractible then the bundle is universal) I am trying to show if the total space of a principal $G$-bundle is contractible, then the bundle is universal. I am across this problem:
$\require{AMScd}\begin{CD}
E@<\simeq<<E\times EG@>\simeq>> EG\\
@VVV@VVV @VVV\\
B@<\simeq<<(E\times EG)/G@>\simeq>> BG
\end{CD}$\
If we have such a commutative diagram, such that $EG\to BG$ is a universal $G$-bundle and $E\times EG\to (E\times EG)/G$, $E\to B$ are principal $G$-bundles with contractible fibers, and the horizontal arrows are homotopy equivalence, then can we show $E\times EG\to (E\times EG)/G$, $E\to B$ are universal $G$-bundles?

For a bundle to universal, you just need the total space to be contractible, if you assume that $E\times EG$ is contactible, so is $E$ therefore, all the bundles designated above are universal.
https://ncatlab.org/nlab/show/universal+principal+bundle
