Theorem ax8dfeq

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

	Ref	Expression
Assertion	ax8dfeq	|- E. z ( ( z e. x -> z e. y ) -> ( w e. x -> w e. y ) )

Proof

Step	Hyp	Ref	Expression
1		ax6e	|- E. z z = w
2		ax8	|- ( w = z -> ( w e. x -> z e. x ) )
3	2	equcoms	|- ( z = w -> ( w e. x -> z e. x ) )
4		ax8	|- ( z = w -> ( z e. y -> w e. y ) )
5	3 4	imim12d	|- ( z = w -> ( ( z e. x -> z e. y ) -> ( w e. x -> w e. y ) ) )
6	1 5	eximii	|- E. z ( ( z e. x -> z e. y ) -> ( w e. x -> w e. y ) )