Theorem absneu 3603

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 3543	. . . . 5 |- ( y = A -> { y } = { A } )
2	1	eqeq2d 2152	. . . 4 |- ( y = A -> ( { x | ph } = { y } <-> { x | ph } = { A } ) )
3	2	spcegv 2777	. . 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 3600	. 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 1332   E.wex 1469   e. wcel 1481   E!weu 2000   {cab 2126   {csn 3532

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-sn 3538

This theorem is referenced by:  rabsneu  3604