Theorem a9evsep 3879

Description: Derive a weakened version of ax-i9 1423, where x and y must be distinct, from Separation ax-sep 3875 and Extensionality ax-ext 2022. The theorem -. A. x -. x = y also holds (ax9vsep 3880), but in intuitionistic logic E. x x = y is stronger. (Contributed by Jim Kingdon, 25-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
a9evsep	|- E. x x = y

Distinct variable group:   x, y

Proof of Theorem a9evsep

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-sep 3875	. 2 |- E. x A. z ( z e. x <-> ( z e. y /\ ( z = z -> z = z ) ) )
2		id 19	. . . . . . . 8 |- ( z = z -> z = z )
3	2	biantru 286	. . . . . . 7 |- ( z e. y <-> ( z e. y /\ ( z = z -> z = z ) ) )
4	3	bibi2i 216	. . . . . 6 |- ( ( z e. x <-> z e. y ) <-> ( z e. x <-> ( z e. y /\ ( z = z -> z = z ) ) ) )
5	4	biimpri 124	. . . . 5 |- ( ( z e. x <-> ( z e. y /\ ( z = z -> z = z ) ) ) -> ( z e. x <-> z e. y ) )
6	5	alimi 1344	. . . 4 |- ( A. z ( z e. x <-> ( z e. y /\ ( z = z -> z = z ) ) ) -> A. z ( z e. x <-> z e. y ) )
7		ax-ext 2022	. . . 4 |- ( A. z ( z e. x <-> z e. y ) -> x = y )
8	6, 7	syl 14	. . 3 |- ( A. z ( z e. x <-> ( z e. y /\ ( z = z -> z = z ) ) ) -> x = y )
9	8	eximi 1491	. 2 |- ( E. x A. z ( z e. x <-> ( z e. y /\ ( z = z -> z = z ) ) ) -> E. x x = y )
10	1, 9	ax-mp 7	1 |- E. x x = y

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   = wceq 1243   E.wex 1381   e. wcel 1393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427  ax-ext 2022  ax-sep 3875

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  ax9vsep  3880