It is commonplace to consider standard axiomatic systems (e.g. $ZF$) with one of the 'less essential' axioms negated, like infinity, 'less essential' here having some ambiguous definition related to an individuals intuition and philosophical preferences.

Has there been any work done exploring what happens if we negate 'essential' axioms?

For example, what about $ZF$ with the axiom of extensionality negated?

My vague understanding of type theories is that axioms do not play such a central role, replaced largely by rules of inference (which are essentially axioms about reasoning instead of the objects of the theory, as far as I can tell? [*cue dunce cap]). But even in this setting, I think it is a coherent question to ask 'what if we take the negation of one of the 'essential' rules of inference as the rule instead?'.

This question is asked mostly as a lark; I am curious what the mathematical universe looks like when we explore a version of it with core pieces of our intuition 'reversed'. Any pointers are appreciated.

In my original edit, I claimed that just negating outright for universal axioms 'wouldn't have any interesting consequences' (which is wrong, see Noah's answer), and also that we might instead negate something like extensionality by claiming that equal sets never have the same elements, which is always wrong (see Wojowu's and Naïm's comments).

alwayshave the same elements. That's a property of equality in classical logic way more fundamental than any axiom. $\endgroup$2more comments