Anti-correlations in the EPR experiment

To measure the a single-qubit observable, you rotate the qubit to align that observable's axis with the computational basis (e.g. align it along the Z axis) then do a measurement. In other words, for any single-qubit observable $O$ you measure $M_O$ by instead performing a single-qubit operation $u$ then a Z-basis measurement $M_Z$. For all $M_O$, there exists a $u$ such that $M_Z \cdot u \equiv M_O$.

Now consider the case where two parties are measuring $M_O$ on their respective parts of a singlet state. They perform $M_O$ by applying $u$ and then measuring $M_Z$. However, the singlet state is not changed by both parties applying $u$, i.e. $(u \otimes u) \cdot (|01\rangle - |10\rangle)$ gives $|01\rangle - |10\rangle$ again (up to global phase). Therefore you can just drop those operations without changing the expected outcome:

$$\begin{align} (M_O \otimes M_O) \cdot (|01\rangle - |10\rangle) &= (M_Z \cdot u) \otimes (M_Z \cdot u) \cdot (|01\rangle - |10\rangle) \\ &= (M_Z \otimes M_Z) \cdot (u \otimes u) \cdot (|01\rangle - |10\rangle) \\ &\propto (M_Z \otimes M_Z) \cdot (|01\rangle - |10\rangle) \\ &\rightarrow \text{measurements give opposite answers} \end{align}$$

Because the singlet state gives opposite answers in the computation basis, and the singlet state is not affected by doing a basis change to both qubits, it must give opposite answers in every basis.