In the first order logic system, on p121 of Ebbinghaus' Mathematical Logic
We call two sets $\Phi$ and $\Psi$ of $S$-sentences equivalent if $Mod^S\Phi = Mod^S\Psi$. Then, in particular, $\Phi \models \phi$ iff $\Phi \models \phi$ for all $\phi \in L^S$.
Are $\Phi$ and $\Psi$ equivalent , if and only if $\Phi \models \phi$ iff $\Psi \models \phi$ for all $\phi \in L^S$? (The last sentence in the quote says only "only if".)
Are two $S$-sentences $\phi$ and $\psi$ equivalent, if and only if $\models (\phi \leftrightarrow \psi)$? (This is used as the definition of logical equivalence between two formulas in propositional logic, on p202.)