Theorem axextdfeq

Description: A version of ax-ext for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010)

	Ref	Expression
Assertion	axextdfeq	$$⊢\exists z\left(\left(z\in x\to z\in y\right)\to \left(\left(z\in y\to z\in x\right)\to \left(x\in w\to y\in w\right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		axextnd	$$⊢\exists z\left(\left(z\in x\leftrightarrow z\in y\right)\to x=y\right)$$
2		ax8	$$⊢x=y\to \left(x\in w\to y\in w\right)$$
3	2	imim2i	$$⊢\left(\left(z\in x\leftrightarrow z\in y\right)\to x=y\right)\to \left(\left(z\in x\leftrightarrow z\in y\right)\to \left(x\in w\to y\in w\right)\right)$$
4	1 3	eximii	$$⊢\exists z\left(\left(z\in x\leftrightarrow z\in y\right)\to \left(x\in w\to y\in w\right)\right)$$
5		biimpexp	$$⊢\left(\left(z\in x\leftrightarrow z\in y\right)\to \left(x\in w\to y\in w\right)\right)\leftrightarrow \left(\left(z\in x\to z\in y\right)\to \left(\left(z\in y\to z\in x\right)\to \left(x\in w\to y\in w\right)\right)\right)$$
6	5	exbii	$$⊢\exists z\left(\left(z\in x\leftrightarrow z\in y\right)\to \left(x\in w\to y\in w\right)\right)\leftrightarrow \exists z\left(\left(z\in x\to z\in y\right)\to \left(\left(z\in y\to z\in x\right)\to \left(x\in w\to y\in w\right)\right)\right)$$
7	4 6	mpbi	$$⊢\exists z\left(\left(z\in x\to z\in y\right)\to \left(\left(z\in y\to z\in x\right)\to \left(x\in w\to y\in w\right)\right)\right)$$