In Predicate Logic Without Equality
The axiom given above assumes that equality is a primitive symbol in predicate logic. Some treatments of axiomatic set theory prefer to do without this, and instead treat the above statement not as an axiom but as a definition of equality. Then it is necessary to include the usual axioms of equality from predicate logic as axioms about this defined symbol. Most of the axioms of equality still follow from the definition; the remaining one is
and it becomes this axiom that is referred to as the axiom of extensionality in this context.
Read more about this topic: Axiom Of Extensionality
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