Theorem ax8dfeq 31704

Description: A version of ax-8 1992 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)

Assertion

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

Proof of Theorem ax8dfeq

Step	Hyp	Ref	Expression
1		ax6e 2250	. 2 |- E. z z = w
2		ax8 1996	. . . 4 |- ( w = z -> ( w e. x -> z e. x ) )
3	2	equcoms 1947	. . 3 |- ( z = w -> ( w e. x -> z e. x ) )
4		ax8 1996	. . 3 |- ( z = w -> ( z e. y -> w e. y ) )
5	3, 4	imim12d 81	. 2 |- ( z = w -> ( ( z e. x -> z e. y ) -> ( w e. x -> w e. y ) ) )
6	1, 5	eximii 1764	1 |- E. z ( ( z e. x -> z e. y ) -> ( w e. x -> w e. y ) )

Syntax hints:   -> wi 4   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)