wrong proof? [because of variable aliasing]

[Jean-Luc-Picard-2021]

I am not 100% sure, but this proof looks wrong:

∀x∀y p(x,y) → ∃x ∃y p(x,y) https://www.umsu.de/trees/#~6x~6yp(x,y)~5~7x~7yp(x,y)

Shouldn't the ∀ rule generate a totally new parameter? In case it is a δ-rule, like here:

First-Order Modal Tableaux https://www.academia.edu/4038862/First_Order_Modal_Tableaux

[Jean-Luc-Picard-2021]

Ok, I can generate the same proof in Fitch style, if I replace in the output generator this here:

Variant 1:

rewrite_term(V, J, K, V) :- var(V), !,
   rewrite_name(J, V), K is J+1.

By this here:

Variant 2:

rewrite_term(V, J, J, V) :- var(V), !,
   rewrite_name(J, V).

These are the two proofs:

Variant 1:

Variant 2:

I think both are valid. But it still scares the shit out of me, that this aliasing still gives a valid proof.