Is there a scenario for when changing the order of different quantifiers in a nested quantifier retain the original meaning? I was exploring the difference in meaning of a proposition when changing the order of two different quantifiers in a nested quantifier. I've been sitting here playing around with various scenarios and doing a bit of research, but I'm unable to come to a decisive answer. Does changing the order of different quantifiers always change the meaning of a logical proposition?
There was two scenarios I was exploring:

*

*Not changing the order of Quantifiers but keeping the order of the variables (e.g. For all x there exists a y ,changed to, For all y there exists an x)


*Changing both the order of the quantifiers and the variables (e.g. For all x there exists a y ,changed to, There exists an x for all y)
If anyone could give me a scenario or a reasoning as to why it's not possible, that would be amazing. Thanks in advance.


Does changing the order of different quantifiers always change the meaning of a logical proposition?

No. Adding to David Lui's example:

e.g. For all x there exists a y, changed to For all y there exists an x

$$∀x∃y(Px → Py) \equiv ∀y∃x(Px → Py);$$

e.g. For all x there exists a y, changed to There exists an x for all y

$$∀x∃y(Px → Py) \equiv ∃x∀y(Px → Py);$$
switching around ∃y and ∀x
$$∀x∃y(Px → Py) \equiv ∃y∀x(Px → Py).$$
