Well this statement isn't exactly true, let me show you why.
take the following array {1,2,-10}
the maximum subarray sum is 1+2=3
but according to your formula its max prefix — min prefix = 3-(-7)=11
which obviously isn't true.
What you're looking for is the maximum value of a prefix sums minus the min prefix sum that ends before it
.

Notation: $n$ length of array, $a_i$ ($0 \le i < n$) $i$th element, $s_{lr} = \sum_{l\le j<r}a_j$ ($0 \le l \le r \le n$) sum of elements from $l$ to $r$, $p_i = s_{0i}$ prefix sum

Part 1: maximum absolute sum of a subarray $\ge$ max prefix sum $-$ min prefix sum

Let $p_M$ and $p_m$ be the maximum and minimum prefix sum, respectively. WLOG assume $M \le m$. Then $\max(|s_{lr}|) \ge |s_{Mm}| = |p_m - p_M| = p_M - p_m$.

Part 2: maximum absolute sum of a subarray $\le$ max prefix sum $-$ min prefix sum

Let $L$, $R$ ($L\le R$) be indices such that $\max(|s_{lr}|) = |s_{LR}|$. WLOG assume $p_L \le p_R$. Then, $\max(|s_{lr}|) = |s_{LR}| = |p_R - p_L| = p_R - p_L \le \max(p_i) - \min(p_i)$.