Proving that the triple integral can be evaluated by iterated integrals

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I am currently in a Calculus III class and we're learning triple integrals. While learning double integrals, I was introduced to Fubini's Theorem, stating the double integral is equal to the iterated integral with respect to $x$ and $y$.

That proof is quite complicated and wasn't covered in my class, but I would love to understand it more once I've taken other more advanced maths. Anyway, the textbook gives a similar manner for evaluating triple integrals by using iterated integrals with respect to $z, y$, and $x$.

Is the proof for this triple integration essentially an expanded version of Fubini's Theorem, or is it another proof altogether? Thank you very much!