A question regarding conditional independence

Question

I have the following random variables: $X= Z + V $ and $ Y $, where $ (V, Y)\perp \!\!\! \perp Z $ but $ Y \not\!\perp\!\!\!\perp V $. Is it true that $ Y \perp \!\!\! \perp X = Z+V |V $? Intuitively, it seems that it is true because given $ V $ all the variation in $ X $ is due to $ Z $, and $ Z $ and $ Y $ are independent.