Is an intersection between NPC and coNPC always DPC? The class DP is defined as the set of all languages $L$ such that $L = L_1 \cap L_2$ where $L_1 \in NP$ and $L_2 \in coNP$.
For example, the language SAT-UNSAT is defined in the following manner ($\varphi$ is a $3-CNF$ formula):
$\{( \varphi, \varphi'): \varphi \in SAT \wedge \varphi ' \in UNSAT \}$. This language is the intersection of:
$$L_1 = \{ (\varphi, \varphi'): \varphi \in SAT \} $$
And:
$$ L_2  = \{ (\varphi, \varphi'): \varphi' \in UNSAT \}. $$

Given $L_1$ which is known to be a NP complete problem, and $L_2$ which is known to be coNP complete problem, does $L_1 \cap L_2$ necessarily DP complete?
(I believe that the answer is yes, because I can use the reduction of $L_1$ and $L_2$. But I am still not very confident in this idea.)
NO, it is not always so.

$L_1=\{&lt;0,\varphi&gt;|\varphi\ \mathrm{is\ satisfiable}\}$

$L_2=\{&lt;1,\varphi&gt;|\varphi\ \mathrm{is\ unsatisfiable}\}$

$L_1\cap L_2=\emptyset$

EDIT: As commented below by David Richerby, $L_1=SAT$ and $L_2=UNSAT$ would likewise falsify your claim in the question.