Let be $E$ a vector space of infinite dimension.
I have the intuition that for all hyperplanes $F$ of $E$, there is an isomorphism between $F$ and $E$, it suffices to show this is true for an hyperplane and this will be true for all due to isomorphism between hyperplanes.
Then, I tried multiple things but I'm not allowed to use existence of basis in infinite dimension vector space or AC explicitly (or any equivalents statements).
Hint: The trick is that we don't need a whole basis, it's enough to pick a countably infinite set of linearly independent vectors in $F$.