Am I using EI right on line 6? (Actually, I'm pretty sure the answer is 'no', and there's a few sketchy lines after that, too. So maybe you could also give a hint about how to do this).
Prove:
(∃x)(∀y)(Gxy → Hxy) (Premise)
(∀x)(∃y)¬Hxy (Premise)
∴ ¬(∀x)(∀y)Gxy
(∀y)(Gay → Hay) (1, Existential Instantiation)
Gab → Hab (3, Universal Instantiation)
(∃y)¬Hay (2, Universal Instantiation)
¬Hab (5, Existential Instantiation)
¬Gab (4, 6, Modus Tolens)
(∃y)¬Gay (7, Existential Generalization)
¬(∀y)Gay (8, Quantifier Negation)
(∃x)¬(∀y)Gxy (9, Existential Generalization)
¬(∀x)(∀y)Gxy (10, Quantifier Negation)