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.)
Is an intersection between NPC and coNPC always DPC?
complexity theory;np complete;complexity classes