Let $T$ be a theory in first order logic over some language $L$. Let $\mathfrak A$ be some structure over $L$ with $\mathfrak A \models T$ and with $A$ be its universe. Then consider every $a \in A$ as a constant and look at the enriched language $L(A) = L \cup A$ with the $L(A)$-structure $\mathfrak A_A = (\mathfrak A, a)_{a\in A}$. A formula over $L(A)$ is called basic if it is an atomic formula. The set $$ \operatorname{Diag}(\mathfrak A) = \{ \varphi \mbox{ is a basic $L(A)$-sentence } \mid \mathfrak A_A \models \varphi \} $$ is called the atomic diagram of $\mathfrak A$.
A theory $T$ is called model-complete if every substructure relation between two models is actually an elementary embedding.
Then $T$ is model-complete if and only if for any $\mathfrak A \models T$ the theory $T \cup \operatorname{Diag}(\mathfrak A)$ is complete.
These definitions are from A course in model theory by K.Tent/M.Zeigler.
I do not understand the quoted statement. For let $t_1 = t_2$ and $t_3 \ne t_4$ be two atomic sentences for terms $t_1, t_2,t_3,t_4$ in $L(A)$. Then Set $\varphi = (t_1 = t_2) \land (t_3 \ne t_4)$. Now suppose in the terms we have some constants from $A$. Then neither $\varphi$ nor $\neg \varphi$ is in $T \cup \operatorname{Diag}(\mathfrak A)$ as it is not in $\operatorname{Diag}(\mathfrak A)$ for it is not atomic, nor is it in $T$ as it is a statement over the enrichted language $L(A)$, but not over $L$. Could someone please explain the above statement (and what I oversee here...)?