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25.6 Quantifier Reorder (QR)

This equivalence deals with multiple quantifiers, which you haven’t learned much about yet. We’ll cover that in Chapter 27.

But since it is related to all the other equivalences in this chapter, we mention it here.

Quantifier Reorder (QR):
AxAyL(x,y) ⇔ AyAxL(x,y)
ExEyL(x,y) ⇔ EyExL(x,y)

Here’s the idea: when you have multiple quantifiers and they are of the same type, then order doesn’t matter. You can reorder them and preserve equivalence.

Foreshadowing: if the quantifiers changing places are a mixture of universal and existential, then reordering violates equivalence.

So even if you have a mixture of different sorts of quantifiers in one sentence, you can reorder them as long as that never means swapping the order of an A and an E.