Theorem axextndbi 31710

Description: axextnd 9413 as a biconditional. (Contributed by Scott Fenton, 14-Dec-2010.)

Assertion

Ref	Expression
axextndbi	|- E. z ( x = y <-> ( z e. x <-> z e. y ) )

Proof of Theorem axextndbi

Step	Hyp	Ref	Expression
1		axextnd 9413	. . 3 |- E. z ( ( z e. x <-> z e. y ) -> x = y )
2		elequ2 2004	. . . 4 |- ( x = y -> ( z e. x <-> z e. y ) )
3	2	jctl 564	. . 3 |- ( ( ( z e. x <-> z e. y ) -> x = y ) -> ( ( x = y -> ( z e. x <-> z e. y ) ) /\ ( ( z e. x <-> z e. y ) -> x = y ) ) )
4	1, 3	eximii 1764	. 2 |- E. z ( ( x = y -> ( z e. x <-> z e. y ) ) /\ ( ( z e. x <-> z e. y ) -> x = y ) )
5		dfbi2 660	. . 3 |- ( ( x = y <-> ( z e. x <-> z e. y ) ) <-> ( ( x = y -> ( z e. x <-> z e. y ) ) /\ ( ( z e. x <-> z e. y ) -> x = y ) ) )
6	5	exbii 1774	. 2 |- ( E. z ( x = y <-> ( z e. x <-> z e. y ) ) <-> E. z ( ( x = y -> ( z e. x <-> z e. y ) ) /\ ( ( z e. x <-> z e. y ) -> x = y ) ) )
7	4, 6	mpbir 221	1 |- E. z ( x = y <-> ( z e. x <-> z e. y ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-cleq 2615  df-clel 2618  df-nfc 2753

This theorem is referenced by: (None)