CFL are not closed under intersection

If L1 and If L2 are two context free languages, their intersection L1 ∩ L2 need not be context free.

For example,

L1 = { aⁿbⁿc^m | n >= 0 and m >= 0 }

L1 says number of a’s should be equal to number of b’s.

L2 = (a^mbⁿcⁿ | n >= 0 and m >= 0 }

L2 says number of b’s should be equal to number of c’s.

L3 = L1 ∩ L2 = { anbncn | n >= 0 }

L1 intersection L2 says both conditions need to be true.

But push down automata can compare only two. So it cannot be accepted by pushdown automata, hence not context free.