In his Senior Thesis, Samuel Coskey answered the question of which axioms of $ZFC$ hold at each stage of the cumulative hierarchy. Here is the list of his results:

Axioms that always hold: Extensionality, Foundation, Union, Axiom Schema of Separation, Choice.

Axioms that hold in $V_{\alpha}$ iff $\alpha$$\gt$0: Empty Set.

Axioms that hold in $V_{\alpha}$ iff $\alpha$$\gt$$\omega$: Infinity.

Axioms that hold at limit ordinals: Power Set, Pairing.

Axioms that hold at inaccessible cardinals: Axiom Schema of Replacement.

My question is simply this:

What changes (if anything) in these results if Choice is dropped?