can you help me to transform ∀x FO logical formula to it equivalent ¬∃ formula?

Question

i have this formula

∀x ∀y (A(x,y) V  A(y,x)  → B(x,c1) ∧ B(y,c2) ∧ c1≠c2)

to the equivalent formula that start by ¬∃x ¬∃ y ?

you will find the question here also

here

Answer 1

The basic equivalence you want to use is: $\forall \equiv \neg\exists\neg$. Thus, $$\begin{align} \forall x \forall y \,\phi(x,y) & \equiv \neg\exists x \neg \, \neg \exists y \neg\,\phi(x,y) \\ & \equiv \neg\exists x \exists y \,\neg\,\phi(x,y). \end{align}$$