Theorem zfcndext

Description: Axiom of Extensionality ax-ext , reproved from conditionless ZFC version and predicate calculus. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 15-Aug-2003) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	zfcndext	$$⊢\forall z\left(z\in x\leftrightarrow z\in y\right)\to x=y$$

Proof

Step	Hyp	Ref	Expression
1		axextnd	$$⊢\exists z\left(\left(z\in x\leftrightarrow z\in y\right)\to x=y\right)$$
2	1	19.36iv	$$⊢\forall z\left(z\in x\leftrightarrow z\in y\right)\to x=y$$