At best an proof-checker can invalidate a purported proof, it absolutely is not going to prove that no proofs are valid (it would have to affirmatively prove that the result is actually independent of the axioms you picked, which is a lot harder)
Many incorrect proofs of this postulate were given over the years. (For instance, in some cases these erroneous proofs have steps that implicitly use the postulate itself.)
The proof checker could detect such errors in a purportedly clean proof.
