Theorem ax6vsep

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

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

Proof

Step	Hyp	Ref	Expression
1		ax-sep	|- E. x A. z ( z e. x <-> ( z e. y /\ ( z = z -> z = z ) ) )
2		id	|- ( z = z -> z = z )
3	2	biantru	|- ( z e. y <-> ( z e. y /\ ( z = z -> z = z ) ) )
4	3	bibi2i	|- ( ( z e. x <-> z e. y ) <-> ( z e. x <-> ( z e. y /\ ( z = z -> z = z ) ) ) )
5	4	biimpri	|- ( ( z e. x <-> ( z e. y /\ ( z = z -> z = z ) ) ) -> ( z e. x <-> z e. y ) )
6	5	alimi	|- ( A. z ( z e. x <-> ( z e. y /\ ( z = z -> z = z ) ) ) -> A. z ( z e. x <-> z e. y ) )
7		ax-ext	|- ( A. z ( z e. x <-> z e. y ) -> x = y )
8	6 7	syl	|- ( A. z ( z e. x <-> ( z e. y /\ ( z = z -> z = z ) ) ) -> x = y )
9	8	eximi	|- ( E. x A. z ( z e. x <-> ( z e. y /\ ( z = z -> z = z ) ) ) -> E. x x = y )
10	1 9	ax-mp	|- E. x x = y
11		df-ex	|- ( E. x x = y <-> -. A. x -. x = y )
12	10 11	mpbi	|- -. A. x -. x = y