∀x(∀z(β)→∃y(¬α)) ⟹∀x(¬∀z(β)∨∃y(¬α)) ⟹¬∃x¬(¬∀z(β)∨∃y(¬α)) ⟹¬∃x(∀z(β)∧¬∃y(¬α)) ⟹¬∃x(∀z(β)∧∀y(α))

In the third line why 2 negations are used ?