Decidability of whether CFL = RL Let L1 be a language generated by a CFG. 
Let L2 be a language generated by a regular grammar.
Is L1 = L2 ?

Is the above problem decidable or undecidable ?



If L1 = L2 then L1 $\cap$ L2' = $\phi$ 

So the above problem can be written as 

CFL $\cap$ RL = $\phi$ or CFL = $\phi$ which is decidable. 

So the answer should be decidable according to me but my professor marked it wrong saying it's undecidable 

Where am I going wrong ?
If $L_1 = L_2$ then $L_1 \cap \overline{L_2} = \emptyset$


While that is true, the reverse is false. The right-hand side is also true if $L_2 \supsetneq L_2$. Therefore, deciding the right-hand side does not solve equality.