First, you have written the axiom of extensionality incorrectly relative to the notion of set theory as a mathematical theory written as a theory written in first-order logic with identity. Compare your sentence to the statements in Jech or Kunen. The conditional direction of your AE is a consequence of the standard account of identity. That account invokes Leibniz' principle of indiscernibility of identicals. The usual axiom of extensionality is the converse, reverse conditional in your statement. You will find that both Jech and Kunen acknowledge this relationship to the theory of identity attached to first-order predicate logic.
Second, your DE is standard object identity. This statement refers to the notion as described in Frege's "Comments on Sense and Reference". To the extent that you associate this notion with properties, you associate sets with concepts. Apparently, this can be a source of contention in some circles. I surmise this from the remarks at FOM found in the link,
There are, apparently, certain idealistic perspectives on sets that are at odds with logicistic conceptualization.
In addition, there is the problem of a predicative ground for the iterative conception of set. In order for the notion of identity in your DE to be definite in the sense of first-order logic with identity, all subclasses of the universe which are sets must be "known". This will be problematic for some views on sets.
First-order model theory circumvents this relationship through use of a denotation schema. That is, for each pair consisting of a constant for the language and a variable of the language one has
'a' denotes x iff a=x
This should be compared with the more familiar
'Snow is white' is true iff snow is white.
By this means, the historical arguments for a predicative ground are preserved relative to the usual axiom of extensionality.
To be clear, the sentence you have labeled AE is the Fregean notion of concept identity. The paradigm of first-order logic with identity separates the indiscernibility of identicals from the identity of indiscernibles. It is only through this distinction that a set can be divorced from conceptual interpretation.
In reply to another question you asked I directed you to the link:
in which Harry Deutsch includes the remark,
"...the false notion that in FOL= the identity symbol defines the
This remark is located in section 5 of the article.
He defines the relation I(A,x,y) in section 1 of the article with the statements,
"Now let A be a set and define the relation I(A,x,y) as follows: For x
and y in A, I(A,x,y) if and only if for each subset X of A, either x
and y are both elements of X or neither is an element of X. This
definition is equivalent to the more usual one identifying the
identity relation on a set A with the set of ordered pairs of the form
for x in A. The present definition proves more helpful in what
It is common for mathematicians to think that identity is eliminable from the language of set theory because the syntax appears to allow it. However, there are ramifications involved because of its impact on the paradigm of first-order logic with identity. Deutsch' article at the Stanford Encyclopedia of Philosophy addresses your question precisely because the arguments for language-relative identity invoke the indiscernibility expressed in your DE.
A small text that can help clarify these matters for anyone interested is "Understanding Identity Statements" by Thomas Morris. Although he presents a theory of regulative identity that would not be of interest to most mathematicians, he precedes that presentation with three chapters on other roles played by identity statements. There is the objectual role that corresponds with the ontological interpretation. There is the metalinguistic role that corresponds with the semantic interpretation (this is the interpretation that treats informative identity such as 'x=y' as stipulative). And, there is the epistemic role that treats informative identity problematically. It is in this latter role where Leibniz' principles warrant the truth or falsity of an informative identity statement.
Well, I expect this reply to be ignored. But, many of your questions are pointing to very non-standard views. You should read Deutsch' article linked above. Also, read the article on Skolem's paradox at the Stanford Encyclopedia of Philosophy. You will find mention of papers by Skolem and Zermelo that you can find in "From Frege to Goedel" by van Heijenoort. It is important to understand the standard paradigm if you are going to entertain contrary views.