The two proofs differ in whether existential elimination is outside or inside negation introduction. For the first proof, choose fresh , derive the negation inside the witness subproof, and then discharge the witness:
(1)
The conclusion at line 6 contains no free . For the second proof, open the universal assumption first, eliminate the existential to derive falsity, and then discharge the universal assumption:
(2)
At line 6 of the second proof, neither the exported conclusion nor the other open assumptions contains free, satisfying the eigenvariable condition. Neither proof needs RAA or double-negation elimination.

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