Theorem ax8dfeq

Description: A version of ax-8 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010)

	Ref	Expression
Assertion	ax8dfeq	$$⊢\exists z\left(\left(z\in x\to z\in y\right)\to \left(w\in x\to w\in y\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ax6e	$$⊢\exists zz=w$$
2		ax8	$$⊢w=z\to \left(w\in x\to z\in x\right)$$
3	2	equcoms	$$⊢z=w\to \left(w\in x\to z\in x\right)$$
4		ax8	$$⊢z=w\to \left(z\in y\to w\in y\right)$$
5	3 4	imim12d	$$⊢z=w\to \left(\left(z\in x\to z\in y\right)\to \left(w\in x\to w\in y\right)\right)$$
6	1 5	eximii	$$⊢\exists z\left(\left(z\in x\to z\in y\right)\to \left(w\in x\to w\in y\right)\right)$$