Ok so my question is as follows; Let A be a substructure of B and S $\subset$ B be a definable set define by a universal formula $\phi(x)$, I need to show that in A $\phi$ defines $A \cap S$ My thought process was as follows, firsly we can define S as $S=\{ b \in B: B \models \phi(b) \}$ if i can show this is a substructure of B and show that $A \cap S$ is a substructure of B (which i think would be easy once i know S is a substructure of B, i can easily deduce what i am trying to prove using the fact (an exercise i just proved) that if a universal formula models B it models all substructures of B.
Any help or hints?