This is the sentence that needs to be transformed: ∃x∀y(Ax → (Bxy ∨ ¬Cy)) → ∀x∃y(Py → Qyx) I have gotten to the point where I eliminated all occurrences of → and imported all negations inside all other logical connectives. Which lead me here:
∃x∀y(Ax ∧ ¬Bxy ∧ Cy) ∨ ∀x∃y(Py ∧¬ Qyx)
But I can't seem to pull the quantifiers in front.
To pull out the quantifiers, we will need to apply some quantifier distributive laws such that $$∀x(Px)∨∀x(Q)↔∀x(Px∨Q)\tag{1}$$ $$∃y(Py)∨∃y(Qy)↔∃y(Py∨Qy)\tag{2}$$ Then we start from \begin{align} &\forall x\exists y(Ax\land\neg Bxy\land Cy)\lor\forall x\exists y(\neg Py \lor Qyx)\\ =&\forall x_1\forall x_2(\exists y(Ax_1\land\neg Bx_1y\land Cy)\lor\exists y(\neg Py \lor Qyx_2))\tag*{By $(1)$}\\ =&\forall x_1\forall x_2\exists y((Ax_1\land\neg Bx_1y\land Cy)\lor(\neg Py \lor Qyx_2))\tag*{By $(2)$}\\ \end{align} Now you can put them into conjunctive normal form.
Note : As @user400188 mentioned, to avoid unnecessary confusion, we could just replace $(1)$ with $$\forall x_1(Px_1)\lor\forall x_2(Qx_2)\leftrightarrow \forall x_1\forall x_2(Px_1\lor Qx_2)$$ or equivalently we can also apply the original $(1)$ twice.