Theorem absneu 3655

Description: Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.)

Assertion

Ref	Expression
absneu	|- ( ( A e. V /\ { x | ph } = { A } ) -> E! x ph )

Proof of Theorem absneu

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 3594	. . . . 5 |- ( y = A -> { y } = { A } )
2	1	eqeq2d 2182	. . . 4 |- ( y = A -> ( { x | ph } = { y } <-> { x | ph } = { A } ) )
3	2	spcegv 2818	. . 3 |- ( A e. V -> ( { x | ph } = { A } -> E. y { x | ph } = { y } ) )
4	3	imp 123	. 2 |- ( ( A e. V /\ { x | ph } = { A } ) -> E. y { x | ph } = { y } )
5		euabsn2 3652	. 2 |- ( E! x ph <-> E. y { x | ph } = { y } )
6	4, 5	sylibr 133	1 |- ( ( A e. V /\ { x | ph } = { A } ) -> E! x ph )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   E.wex 1485   E!weu 2019   e. wcel 2141   {cab 2156   {csn 3583

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-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sn 3589

This theorem is referenced by:  rabsneu  3656