The answer by [Macro](https://stats.stackexchange.com/users/4856/macro) is great, but here is an even simpler way that does not require you to use any outside theorem asserting the conditional distribution.  It involves writing the Mahanalobis distance in a form that separates the argument variable for the conditioning statement, and then factorising the normal density accordingly.

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**Rewriting the Mahanalobis distance for a conditional vector:** This derivation uses a matrix inversion formula that uses the [Schur complement](https://en.wikipedia.org/wiki/Schur_complement) $\boldsymbol{\Sigma}_* \equiv \boldsymbol{\Sigma}_{11} - \boldsymbol{\Sigma}_{12} \boldsymbol{\Sigma}_{22}^{-1} \boldsymbol{\Sigma}_{21}$.  We first use the [blockwise inversion formula](https://en.wikipedia.org/wiki/Invertible_matrix#Blockwise_inversion) to write the inverse-variance matrix as:

$$\begin{equation} \begin{aligned}
\boldsymbol{\Sigma}^{-1} 
= \begin{bmatrix} 
\boldsymbol{\Sigma}_{11} & \boldsymbol{\Sigma}_{12} \\
\boldsymbol{\Sigma}_{21} & \boldsymbol{\Sigma}_{22} \\
\end{bmatrix}^{-1} 
= \begin{bmatrix} 
\boldsymbol{\Sigma}_{11}^* & \boldsymbol{\Sigma}_{12}^* \\
\boldsymbol{\Sigma}_{21}^* & \boldsymbol{\Sigma}_{22}^* \\
\end{bmatrix}, 
\end{aligned} \end{equation}$$

where:

$$\begin{equation} \begin{aligned}
\begin{matrix} 
\boldsymbol{\Sigma}_{11}^* = \boldsymbol{\Sigma}_*^{-1} \text{ } \quad \quad \quad \quad & & & & & \boldsymbol{\Sigma}_{12}^* = -\boldsymbol{\Sigma}_*^{-1} \boldsymbol{\Sigma}_{12} \boldsymbol{\Sigma}_{22}^{-1}, \quad \quad \quad \\[6pt]
\boldsymbol{\Sigma}_{21}^* = - \boldsymbol{\Sigma}_{22}^{-1} \boldsymbol{\Sigma}_{12} \boldsymbol{\Sigma}_*^{-1} & & & & & \boldsymbol{\Sigma}_{22}^* = \boldsymbol{\Sigma}_{22}^{-1} + \boldsymbol{\Sigma}_{22}^{-1} \boldsymbol{\Sigma}_{12} \boldsymbol{\Sigma}_*^{-1} \boldsymbol{\Sigma}_{12} \boldsymbol{\Sigma}_{22}^{-1}. \text{ } \\[6pt]
\end{matrix} 
\end{aligned} \end{equation}$$

Using this formula we can now write the Mahanalobis distance as:

$$\begin{equation} \begin{aligned}
(\boldsymbol{y} &- \boldsymbol{\mu})^\text{T} \boldsymbol{\Sigma}^{-1} (\boldsymbol{y} - \boldsymbol{\mu}) \\[6pt]
&= \begin{bmatrix} \boldsymbol{y}_1 - \boldsymbol{\mu}_1 \\ \boldsymbol{y}_2 - \boldsymbol{\mu}_2 \end{bmatrix}^\text{T} \begin{bmatrix} 
\boldsymbol{\Sigma}_{11}^* & \boldsymbol{\Sigma}_{12}^* \\
\boldsymbol{\Sigma}_{21}^* & \boldsymbol{\Sigma}_{22}^* \\
\end{bmatrix} \begin{bmatrix} \boldsymbol{y}_1 - \boldsymbol{\mu}_1 \\ \boldsymbol{y}_2 - \boldsymbol{\mu}_2 \end{bmatrix} \\[6pt]
&= \quad (\boldsymbol{y}_1 - \boldsymbol{\mu}_1)^\text{T} \boldsymbol{\Sigma}_{11}^* (\boldsymbol{y}_1 - \boldsymbol{\mu}_1)
+ (\boldsymbol{y}_1 - \boldsymbol{\mu}_1)^\text{T} \boldsymbol{\Sigma}_{12}^* (\boldsymbol{y}_2 - \boldsymbol{\mu}_2) \\[6pt]
&\quad + (\boldsymbol{y}_2 - \boldsymbol{\mu}_2)^\text{T} \boldsymbol{\Sigma}_{21}^* (\boldsymbol{y}_1 - \boldsymbol{\mu}_1)
+ (\boldsymbol{y}_2 - \boldsymbol{\mu}_2)^\text{T} \boldsymbol{\Sigma}_{22}^* (\boldsymbol{y}_2 - \boldsymbol{\mu}_2) \\[6pt]
&= \quad (\boldsymbol{y}_1 - \boldsymbol{\mu}_1)^\text{T} \boldsymbol{\Sigma}_\text{S}^{-1} (\boldsymbol{y}_1 - \boldsymbol{\mu}_1)
- (\boldsymbol{y}_1 - \boldsymbol{\mu}_1)^\text{T} \boldsymbol{\Sigma}_\text{S}^{-1} \boldsymbol{\Sigma}_{12} \boldsymbol{\Sigma}_{22}^{-1} (\boldsymbol{y}_2 - \boldsymbol{\mu}_2) \\[6pt]
&\quad - (\boldsymbol{y}_2 - \boldsymbol{\mu}_2)^\text{T} \boldsymbol{\Sigma}_{22}^{-1} \boldsymbol{\Sigma}_{12} \boldsymbol{\Sigma}_\text{S}^{-1} (\boldsymbol{y}_1 - \boldsymbol{\mu}_1)
+ (\boldsymbol{y}_2 - \boldsymbol{\mu}_2)^\text{T} \boldsymbol{\Sigma}_{22}^{-1} (\boldsymbol{y}_2 - \boldsymbol{\mu}_2) \\[6pt]
&\quad + (\boldsymbol{y}_2 - \boldsymbol{\mu}_2)^\text{T} \boldsymbol{\Sigma}_{22}^{-1} \boldsymbol{\Sigma}_{12} \boldsymbol{\Sigma}_\text{S}^{-1} \boldsymbol{\Sigma}_{12} \boldsymbol{\Sigma}_{22}^{-1} (\boldsymbol{y}_2 - \boldsymbol{\mu}_2) \\[6pt]
&= (\boldsymbol{y}_1 - (\boldsymbol{\mu}_1 + \boldsymbol{\Sigma}_{12} \boldsymbol{\Sigma}_{22}^{-1} (\boldsymbol{y}_2 - \boldsymbol{\mu}_2)))^\text{T} \boldsymbol{\Sigma}_\text{S}^{-1} (\boldsymbol{y}_1 - (\boldsymbol{\mu}_1 + \boldsymbol{\Sigma}_{12} \boldsymbol{\Sigma}_{22}^{-1} (\boldsymbol{y}_2 - \boldsymbol{\mu}_2))) \\[6pt]
&\quad + (\boldsymbol{y}_2 - \boldsymbol{\mu}_2)^\text{T} \boldsymbol{\Sigma}_{22}^{-1} (\boldsymbol{y}_2 - \boldsymbol{\mu}_2) \\[6pt]
&= (\boldsymbol{y}_1 - \boldsymbol{\mu}_*)^\text{T} \boldsymbol{\Sigma}_*^{-1} (\boldsymbol{y}_1 - \boldsymbol{\mu}_*) + (\boldsymbol{y}_2 - \boldsymbol{\mu}_2)^\text{T} \boldsymbol{\Sigma}_{22}^{-1} (\boldsymbol{y}_2 - \boldsymbol{\mu}_2) , \\[6pt]
\end{aligned} \end{equation}$$

where $\boldsymbol{\mu}_* \equiv \boldsymbol{\mu}_1 + \boldsymbol{\Sigma}_{12} \boldsymbol{\Sigma}_{22}^{-1} (\boldsymbol{y}_2 - \boldsymbol{\mu}_2)$ is the **adjusted mean vector**.  Note that this result is a general result that does not assume normality of the random vectors.  It gives a useful way of decomposing the Mahanalobis distance so that it consists of a sum of quadratic forms on the marginal and conditional parts.  In the conditional part the conditioning vector $\boldsymbol{y}_2$ is absorbed into the mean vector and variance matrix.  To clarify the form, we repeat the equation with labelling of terms:

$$(\boldsymbol{y} - \boldsymbol{\mu})^\text{T} \boldsymbol{\Sigma}^{-1} (\boldsymbol{y} - \boldsymbol{\mu}) 
= \underbrace{(\boldsymbol{y}_1 - \boldsymbol{\mu}_*)^\text{T} \boldsymbol{\Sigma}_*^{-1} (\boldsymbol{y}_1 - \boldsymbol{\mu}_*)}_\text{Conditional Part} + \underbrace{(\boldsymbol{y}_2 - \boldsymbol{\mu}_2)^\text{T} \boldsymbol{\Sigma}_{22}^{-1} (\boldsymbol{y}_2 - \boldsymbol{\mu}_2)}_\text{Marginal Part}.$$

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**Deriving the conditional distribution:** Now that we have the above form for the Mahanalobis distance, the rest is easy.  We have:

$$\begin{equation} \begin{aligned}
p(\boldsymbol{y}_1 | \boldsymbol{y}_2, \boldsymbol{\mu}, \boldsymbol{\Sigma})
&\overset{\boldsymbol{y}_1}{\propto} p(\boldsymbol{y}_1 , \boldsymbol{y}_2 | \boldsymbol{\mu}, \boldsymbol{\Sigma}) \\[12pt]
&= \text{N}(\boldsymbol{y} | \boldsymbol{\mu}, \boldsymbol{\Sigma}) \\[10pt]
&\overset{\boldsymbol{y}_1}{\propto} \exp \Big( - \frac{1}{2} (\boldsymbol{y} - \boldsymbol{\mu})^\text{T} \boldsymbol{\Sigma}^{-1} (\boldsymbol{y} - \boldsymbol{\mu}) \Big) \\[6pt]
&\overset{\boldsymbol{y}_1}{\propto} \exp \Big( - \frac{1}{2} (\boldsymbol{y}_1 - \boldsymbol{\mu}_*)^\text{T} \boldsymbol{\Sigma}_*^{-1} (\boldsymbol{y}_1 - \boldsymbol{\mu}_*) \Big) \\[6pt]
&\overset{\boldsymbol{y}_1}{\propto}\text{N}(\boldsymbol{y}_1 | \boldsymbol{\mu}_*, \boldsymbol{\Sigma}_*). \\[6pt]
\end{aligned} \end{equation}$$

This establishes that the conditional distribution is also multivariate normal, with the specified conditional mean vector and conditional variance matrix.