query: is there a difference between the original and the following modified axiom of extensionality?
positive_passages.docid: 147908
positive_passages.text: Why use the biconditional in the Axiom of Extensionality
negative_passages.docid: 157490
negative_passages.text: Statements equivalent to the Axiom of Choice
negative_passages.docid: 130998
negative_passages.text: Is $\exists x \in A ~:~ P(x)$ the same as $\exists x ~:~ x \in A \implies P(x)$
negative_passages.docid: 39755
negative_passages.text: In a faithfully flat ring extension, is $\operatorname{ht}I=\operatorname{ht}IS$ right?
negative_passages.docid: 12449
negative_passages.text: How to formulate continuum hypothesis without the axiom of choice?
negative_passages.docid: 142315
negative_passages.text: How does the Axiom of Regularity apply to $A=\{1,2,3\}$?
negative_passages.docid: 142219
negative_passages.text: Does the splitting lemma hold without the axiom of choice?
negative_passages.docid: 77546
negative_passages.text: Does a "backwards" choice function imply the Axiom of Choice?
negative_passages.docid: 48820
negative_passages.text: Where is axiom of regularity actually used?
negative_passages.docid: 104354
negative_passages.text: What is the relation between normal extension and separable extension?
negative_passages.docid: 135222
negative_passages.text: What are the consequences if Axiom of Infinity is negated?
negative_passages.docid: 84485
negative_passages.text: What is the purpose of the Axiom of regularity/foundation?
negative_passages.docid: 744
negative_passages.text: Extendible equals sign
negative_passages.docid: 155207
negative_passages.text: Confusion regarding one formulation of the Axiom of Choice.
negative_passages.docid: 58631
negative_passages.text: Continuity and the Axiom of Choice
negative_passages.docid: 112125
negative_passages.text: In an extension field, is there any difference between the original field and its isomorphic copy in the extension field?
negative_passages.docid: 109039
negative_passages.text: Does this "extension property" for polynomial rings satisfy a universal property?