Theorem euabsn

Description: Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by NM, 22-Feb-2004)

	Ref	Expression
Assertion	euabsn	|- ( E! x ph <-> E. x { x | ph } = { x } )

Proof

Step	Hyp	Ref	Expression
1		euabsn2	|- ( E! x ph <-> E. y { x | ph } = { y } )
2		nfv	|- F/ y { x | ph } = { x }
3		nfab1	|- F/_ x { x | ph }
4	3	nfeq1	|- F/ x { x | ph } = { y }
5		sneq	|- ( x = y -> { x } = { y } )
6	5	eqeq2d	|- ( x = y -> ( { x | ph } = { x } <-> { x | ph } = { y } ) )
7	2 4 6	cbvexv1	|- ( E. x { x | ph } = { x } <-> E. y { x | ph } = { y } )
8	1 7	bitr4i	|- ( E! x ph <-> E. x { x | ph } = { x } )