So, my understanding of heat capacity at constant volume is that

$$C_V = \bigg(\frac{\delta Q}{dT}\bigg)_V$$

So far, so good. But according to my professor,

$$dU = C_VdT$$

for an ideal gas, even when work is performed, that is, when $\delta Q\neq dU$. I see that there are a lot of explanations in this forum as to why the above formula is true, so please note that I am NOT asking that again. What I am asking for is some help understanding how can $C_V$ be equal to two different things at the same time... That looks like a contradiction. Any help would be appreciated!