From the formula $$\forall f\colon A\to A\,\exists x\colon A\,(f(x)=x)$$ we can get the scheme $$\forall x\colon A\,(\phi(x)\vee\neg\phi(x))\Rightarrow\forall x\colon A\,\phi(x)\vee\forall x\colon A\,\neg\phi(x)$$ From the scheme $$\forall x\colon A\,\exists y\colon A\,\phi(x,y)\Rightarrow\exists x\colon A\,\phi(x,x)$$ we can get $$\forall x\colon A\,(\phi(x)\vee\psi(x))\Rightarrow\neg\neg(\forall x\colon A\,\phi(x)\vee\forall x\colon A\,\psi(x))$$ Where can I find these results? Give me a link, please.

P.S. Sorry, the second scheme must be $$\forall x\colon A\,(\neg\phi(x)\vee\neg\psi(x))\Rightarrow\neg\neg(\forall x\colon A\,\neg\phi(x)\vee\forall x\colon A\,\neg\psi(x))$$ The results are not quite difficult, but I never saw it in articles or books.