I am trying to prove Los theorem (the one for ultraproducts, the statement with a proof is here. https://proofwiki.org/wiki/Łoś%27s_Theorem), but using $\forall$ as primitive quantifier. Any proof online uses $\exists$ as primitive quantifier to do the induction step. Could someone please give a proof using $\forall$ instead of $\exists$ to prove it?
I know, the $\forall$ is just $\lnot \exists \lnot$, but now I am searching for a direct proof. I did not expect it to be hard (basically "by definition"). However, after tried manipulating the statement, I still cannot get it. Any help or reference would be appreciated, thank you!