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?
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)