Theorem ax8dfeq 30493

Description: A version of ax-8 1899 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 2104	. 2 |- E. z z = w
2		ax8 1903	. . . 4 |- ( w = z -> ( w e. x -> z e. x ) )
3	2	equcoms 1874	. . 3 |- ( z = w -> ( w e. x -> z e. x ) )
4		ax8 1903	. . 3 |- ( z = w -> ( z e. y -> w e. y ) )
5	3, 4	imim12d 77	. 2 |- ( z = w -> ( ( z e. x -> z e. y ) -> ( w e. x -> w e. y ) ) )
6	1, 5	eximii 1719	1 |- E. z ( ( z e. x -> z e. y ) -> ( w e. x -> w e. y ) )

Syntax hints:   -> wi 4   E.wex 1673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-12 1943  ax-13 2101

This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674

This theorem is referenced by: (None)