Theorem a9evsep 4104

Description: Derive a weakened version of ax-i9 1518, where x and y must be distinct, from Separation ax-sep 4100 and Extensionality ax-ext 2147. The theorem -. A. x -. x = y also holds (ax9vsep 4105), 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 4100	. 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 300	. . . . . . 7 |- ( z e. y <-> ( z e. y /\ ( z = z -> z = z ) ) )
4	3	bibi2i 226	. . . . . 6 |- ( ( z e. x <-> z e. y ) <-> ( z e. x <-> ( z e. y /\ ( z = z -> z = z ) ) ) )
5	4	biimpri 132	. . . . 5 |- ( ( z e. x <-> ( z e. y /\ ( z = z -> z = z ) ) ) -> ( z e. x <-> z e. y ) )
6	5	alimi 1443	. . . 4 |- ( A. z ( z e. x <-> ( z e. y /\ ( z = z -> z = z ) ) ) -> A. z ( z e. x <-> z e. y ) )
7		ax-ext 2147	. . . 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 1588	. 2 |- ( E. x A. z ( z e. x <-> ( z e. y /\ ( z = z -> z = z ) ) ) -> E. x x = y )
10	1, 9	ax-mp 5	1 |- E. x x = y

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1341   = wceq 1343   E.wex 1480   e. wcel 2136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-ial 1522  ax-ext 2147  ax-sep 4100

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  ax9vsep  4105