Theorem axextg

Description: A generalization of the axiom of extensionality in which x and y need not be distinct. (Contributed by NM, 15-Sep-1993) (Proof shortened by Andrew Salmon, 12-Aug-2011) Remove dependencies on ax-10 , ax-12 , ax-13 . (Revised by BJ, 12-Jul-2019) (Revised by Wolf Lammen, 9-Dec-2019)

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

Proof

Step	Hyp	Ref	Expression
1		elequ2	$$⊢w=x\to \left(z\in w\leftrightarrow z\in x\right)$$
2	1	bibi1d	$$⊢w=x\to \left(\left(z\in w\leftrightarrow z\in y\right)\leftrightarrow \left(z\in x\leftrightarrow z\in y\right)\right)$$
3	2	albidv	$$⊢w=x\to \left(\forall z\left(z\in w\leftrightarrow z\in y\right)\leftrightarrow \forall z\left(z\in x\leftrightarrow z\in y\right)\right)$$
4		equequ1	$$⊢w=x\to \left(w=y\leftrightarrow x=y\right)$$
5	3 4	imbi12d	$$⊢w=x\to \left(\left(\forall z\left(z\in w\leftrightarrow z\in y\right)\to w=y\right)\leftrightarrow \left(\forall z\left(z\in x\leftrightarrow z\in y\right)\to x=y\right)\right)$$
6		ax-ext	$$⊢\forall z\left(z\in w\leftrightarrow z\in y\right)\to w=y$$
7	5 6	chvarvv	$$⊢\forall z\left(z\in x\leftrightarrow z\in y\right)\to x=y$$