Bipartite graph cover problem 
Let $G$ be bipartite with bipartition $A$, $B$. Suppose that $C$ and $C'$ are both covers of $G$. Prove that $C^{\wedge}$ = $(A \cap C \cap C') \cup (B \cap (C \cup C'))$ is also a cover of $G$.

Does anyone know which theorem is useful for proving this statement?

This statement is fairly easy to prove without appealing to any special theorems. 
It might be useful to rewrite the statement

Every edge in $G$ has an endvertex in $C''=(A\cap C\cap C')\cup (B\cap(C\cup C')$,

which you are trying to prove, as the equivalent statement

If an edge $e\in E(G)$ has no endvertex in $B\cap (C\cup C')$, then it has an endvertex in $A\cap C\cap C'$.

Hint: every edge $e$ has an endvertex in $A$, an endvertex in $B$, and at least one endvertex in each of $C$ and $C'$.
Hope this helps!
