Theorem absneu 3648

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 3587	. . . . 5 |- ( y = A -> { y } = { A } )
2	1	eqeq2d 2177	. . . 4 |- ( y = A -> ( { x | ph } = { y } <-> { x | ph } = { A } ) )
3	2	spcegv 2814	. . 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 3645	. 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 1343   E.wex 1480   E!weu 2014   e. wcel 2136   {cab 2151   {csn 3576

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sn 3582

This theorem is referenced by:  rabsneu  3649