The displayed derivation is invalid, although its final conjecture is valid. Its existential-elimination step exports with the local witness name still free, violating the eigenvariable condition. Introducing an existential quantifier afterwards does not repair that earlier invalid rule application.
Move the existential quantifier introduction rule inside the witness scope:
LineFormulaJustification
1Premise
2Premise
3Assume with fresh
4Universal quantifier elimination rule, 1
5Implication elimination rule, 3 and 4
6Existential quantifier introduction rule, 5
7Existential quantifier elimination rule, 2 and 3–6
The exported formula contains no free , and neither original premise contains that name. The witness assumption is now discharged legally.

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