Suppose the day after tomorrow is NOT Saturday. Which of the following days CANNOT be the day before yesterday?

Let the current day be $T$ (Today).
* The day after tomorrow is $T+2$.
* The day before yesterday is $T-2$.

We are given that the day after tomorrow is not Saturday:

$$T+2\ne \text{Saturday}$$

Subtracting $4$ days from both sides tells us about the day before yesterday ($T-2$):

$$T-2\ne \text{Saturday}-4\text{ days}$$

Counting backwards $4$ days from Saturday:
1 day before: Friday
2 days before: Thursday
3 days before: Wednesday
4 days before: Tuesday

Therefore:

$$T-2\ne \text{Tuesday}$$

This means the day before yesterday CANNOT be Tuesday.