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Trying to rectify this formula. I'm having problems eliminating the correct symbols.

$$F = ∀x[∀y(P(x,y)∪Q(x,z)) ∩∃z∃x∀y(¬P(x,y) ∪ ∀z¬Q(x,z))]$$

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$$F = ∀x[∀y(P(x,y)∪Q(x,z))∩∃z∃x∀y(¬P(x,y)∪∀z¬Q(x,z))]$$ Notice that $∃z$ is not needed. $$F = ∀x[∀y(P(x,y)∪Q(x,z))∩∃x1∀y1(¬P(x1,y1) ∪ ∀z1¬Q(x1,z1))]$$

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