Theorem ax6vsep 4785

Description: Derive ax6v 1889 (a weakened version of ax-6 1888 where x and y must be distinct), from Separation ax-sep 4781 and Extensionality ax-ext 2602. See ax6 2251 for the derivation of ax-6 1888 from ax6v 1889. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax6vsep	|- -. A. x -. x = y

Distinct variable group:   x, y

Proof of Theorem ax6vsep

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-sep 4781	. . 3 |- E. x A. z ( z e. x <-> ( z e. y /\ ( z = z -> z = z ) ) )
2		id 22	. . . . . . . . 9 |- ( z = z -> z = z )
3	2	biantru 526	. . . . . . . 8 |- ( z e. y <-> ( z e. y /\ ( z = z -> z = z ) ) )
4	3	bibi2i 327	. . . . . . 7 |- ( ( z e. x <-> z e. y ) <-> ( z e. x <-> ( z e. y /\ ( z = z -> z = z ) ) ) )
5	4	biimpri 218	. . . . . 6 |- ( ( z e. x <-> ( z e. y /\ ( z = z -> z = z ) ) ) -> ( z e. x <-> z e. y ) )
6	5	alimi 1739	. . . . 5 |- ( A. z ( z e. x <-> ( z e. y /\ ( z = z -> z = z ) ) ) -> A. z ( z e. x <-> z e. y ) )
7		ax-ext 2602	. . . . 5 |- ( A. z ( z e. x <-> z e. y ) -> x = y )
8	6, 7	syl 17	. . . 4 |- ( A. z ( z e. x <-> ( z e. y /\ ( z = z -> z = z ) ) ) -> x = y )
9	8	eximi 1762	. . 3 |- ( E. x A. z ( z e. x <-> ( z e. y /\ ( z = z -> z = z ) ) ) -> E. x x = y )
10	1, 9	ax-mp 5	. 2 |- E. x x = y
11		df-ex 1705	. 2 |- ( E. x x = y <-> -. A. x -. x = y )
12	10, 11	mpbi 220	1 |- -. A. x -. x = y

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990

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-ext 2602  ax-sep 4781

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

This theorem is referenced by: (None)