Joint distribution of two brownian bridges

Question

Consider $X_t=B_t-tB_1$ for $0\leq t\leq 1$, where $B_t$ represents the standard Brownian motion. I derived that $E(X_t)=0$ and $\operatorname{Cov}(X_t,X_s)=\min(t,s)-st$. Now, I am asked the joint distribution of $X_t$ and $X_s$ for some fixed $s,t\in[0,1]$. My first thought was that the joint distribution is the bivariate normal with mean $\begin{pmatrix} 0 \\ 0 \end{pmatrix}$ and covariance matrix $\Sigma = \begin{pmatrix} t(1-t) & \min(t,s)-st \\ \min(t,s)-st & s(1-s) \end{pmatrix}$. However, I do not know how to justify this choice. Is it the bivariate normal?