I need a little help with the proof of the validity of the first quantifier rule. So let $F \rightarrow G$ is valid in a structure $S$. I must prove that $\exists F(x) \rightarrow G$ is valid in $S$. Let $S$ be a $L$-structure. Suppose $F \rightarrow G$ is valid in $S$. For any term $t(x,y)$. So $t^S:S^{n+1} \rightarrow S$ and $(a,b) \rightarrow t^S(a,b) \in S$. So must show that $F^S(a,b)$ holds, then $G^S(a,b)$ holds. So i think I first should assume that $F^S(a,b)$ holds. Then should $(\exists F)^S(a,b)$ holds. I need to use the fact that $x$ is not free in $G$. Then use the lemma that says that if $x_i$ does not have a free occurrence in the formula $F(x_1,...,x_i,...,x_n)$ and $(a_1,...,a_i,...,a_n)\in F^S$ then, for any $a_i' \in S$, we have $(a_1,...,a_i',...,a_n)\in F^S$
Thanks for the help