Let abbreviate the given negated existential statement, and choose a fresh variable that is not free in the undischarged premise. Work from an assumed to :
1  ¬∃x (Q(x) → P(x))                  premise N
2  | P(u)                             assume
3  | | Q(u)                           assume
4  | | P(u)                           available from 2
5  | Q(u) → P(u)                      →-intro, discharge 3–4
6  | ∃x (Q(x) → P(x))                 ∃-intro, witness u in 5
7  | ⊥                                ¬-elim, 6 and 1
8  | Q(u)                             ⊥-elim, 7
9  P(u) → Q(u)                        →-intro, discharge 2–8
10 ∀x (P(x) → Q(x))                   ∀-intro, 9, u fresh
The occurrence of in the temporary assumption at line 2 is harmless because that assumption is discharged before universal quantifier introduction rule. No unproved implication-creation rule or quantifier-negation equivalence is needed.

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