Theorem axextb

Description: A bidirectional version of the axiom of extensionality. Although this theorem looks like a definition of equality, it requires the axiom of extensionality for its proof under our axiomatization. See the comments for ax-ext and df-cleq . (Contributed by NM, 14-Nov-2008)

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

Proof

Step	Hyp	Ref	Expression
1		elequ2g	$$⊢x=y\to \forall z\left(z\in x\leftrightarrow z\in y\right)$$
2		axextg	$$⊢\forall z\left(z\in x\leftrightarrow z\in y\right)\to x=y$$
3	1 2	impbii	$$⊢x=y\leftrightarrow \forall z\left(z\in x\leftrightarrow z\in y\right)$$