Isomorphism theorems to conclude that cokernel is finite dimensional

Question

Are the following implications correct: let $U,V$ be two subspaces of $H$ then $$ \frac{H}{U} = \frac{U+V}{U} \cong \frac{V}{U \cap V} \text{ and } \frac{H}{V} \cong \frac{H}{U \cap V } \Big/ { \frac{V}{U \cap V} \cong \frac{H}{U \cap V}} \Big/ \frac{H}{U}. $$ Is this enough to show that if $U$ and $V$ have finite codimension, so has $U \cap V$? Why does this not imply $\operatorname{codim}U+\operatorname{codim}V=\operatorname{codim}(U \cap V)$?