Let $L$ be a first order language and $M$ an interpretation of $L$.
If $A$ and $B$ are two substructures of $M$ and their intersection $C=A\cap B\ne \emptyset$, then is it the case that every sentence (in language $L$) true for both $A$ and $B$ is also true for $C$?


By "intersection" of $A$ and $B$, I mean the smaller substructure whose domain is the intersection of the domain of $A$ and $B$.


I came to this conjecture when I learned that there was a **smallest standard submodel** in a universe of some set theories (such as $ZF$) if there was a standard submodel in it. The **smallest standard submodel** is the intersection of all the standard submodels. But is the intersection of any two standard submodels still a standard submodel? If so, considered every standard submodel is nothing more than a substructure of the universe for which a group of sentences, i.e. the axioms of some set theory, are true, can we further generalize it into the conjecture I mentioned above?


I haven't found a proof or disproof for this so far (actually I am not very good at it). Can anyone prove it or give a counterexample?


-----update----


The question in the title is trivial. Thanks for answers from Andrej Bauer and 喻良. But whether the intersection of two standard submodels of $ZF$ is still a stanadard submodel? (which is my earliest concern and let's only focus on $ZF$ or $ZFC$ here)



Is a sentence true for two substructures also true for their intersection?
If $\mathcal C$ is any substructure of a structure $\mathcal A$, then $\mathcal C$ is the intersection of $\mathcal A$ and an isomorphic copy of it, $\mathcal A'$, inside a larger structure $\mathcal M$. (Take an isomorphic copy of $\mathcal A$ and combine it with $\mathcal A$, identifying corresponding elements of $\mathcal C$.) So a sentence is preserved by intersections if and only if it's preserved by substructures. And that's well-known to be the case only for sentences equivalent to universal sentences.


