Let $\Gamma$ be the set of sentences (formulas without free variables) of the signature $\Sigma $ such that for any algebraic system $\mathfrak{M}$ of signature $\Sigma $ there exists a sentence $ \Phi \in \Gamma $ that is true on $\mathfrak{M}$.
Show that there exists a finite set $\left \{ \Phi _{1}, \Phi _{2}, ...... \Phi _{n} \right \} \subseteq \Gamma $ such that the sentence $\left ( \Phi _{1} \vee \left (\Phi _{2} \vee .... \vee \Phi _{n} \right) ...\right )$ is Is identically true formula.
It sounds intuitively clear, like we have a lot of suggestions and there will always be a true one on some system. but how to show that the true is in a finite set of sentences
Argue by compactness: If what you want to prove is false, $\{\lnot\phi: \phi\in\Gamma\} $ would have a model (contradicting your hypothesis), because each finite subtheory does.