Compute $\mathbb P(\sup_{t\in [a,b]}B_t>x)$, what's wrong with Markov property here?

Question

I would like to compute $\mathbb P\left(\sup_{t\in [a,b]}B_t>x\right)$ where $(B_t)$ is a Brownian motion and $0<a<b$. What I would say using Markov property is $$\mathbb P\left(\sup_{t\in [a,b]}B_t>x \right)=\mathbb P\left(\sup_{t\in [0,b-a]}B_t>x\mid B_0=B_a\right),$$ but something looks strange since the LHS is a number whereas the RHS is a random variable. Can someone tel me how to manage ?