In this "Delayed choice eraser" experiment:
if one looks at all the photons detected by $D_0$ regardless of their coincidence with the other detectors, one sees no interference pattern.
It is obvious by taking the sum of coincidence patterns $\sum_1^4<D_0,D_i>$, since the $<D_0,D_1>$ and $<D_0,D_2>$ patterns have a phase difference of $\pi$, and the $<D_0,D_3>$ and $<D_0,D_4>$ don't show any interference to begin with.
So not looking at all at what the idler photon does, and only looking at $D_0$ we would see no interference.
But without the BBO after the 2 slits, we'd see Young's double slit fringes.
My question is: Why does the BBO destroy the interference pattern from the double slits.