Proof of Demorgan's law
In order to prove $A = B$, It is sufficient to prove that $A'B = 0$ and $A' + B = 1$. Try to think of why this should be the case intuitively. In case you are unable to understand, then think of $A$ and $B$ as sets, Boolean $+$ operation as set union operation and Boolean $.$ operation as set intersection operation.

Therefore, take $A = (X+Y)'$ and $B = X'Y'$.

So, $A'B = (X+Y)X'Y' = XX'Y' + YX'Y' = 0$.

Also, $A' + B = X + Y + X'Y' = 1$.

(Use the property $A + BC = (A + B )(A + C)$ for simplifying above expression.