Theorem absneu 3442

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 3386	. . . . 5 |- ( y = A -> { y } = { A } )
2	1	eqeq2d 2051	. . . 4 |- ( y = A -> ( { x | ph } = { y } <-> { x | ph } = { A } ) )
3	2	spcegv 2641	. . 3 |- ( A e. V -> ( { x | ph } = { A } -> E. y { x | ph } = { y } ) )
4	3	imp 115	. 2 |- ( ( A e. V /\ { x | ph } = { A } ) -> E. y { x | ph } = { y } )
5		euabsn2 3439	. 2 |- ( E! x ph <-> E. y { x | ph } = { y } )
6	4, 5	sylibr 137	1 |- ( ( A e. V /\ { x | ph } = { A } ) -> E! x ph )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   E.wex 1381   e. wcel 1393   E!weu 1900   {cab 2026   {csn 3375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sn 3381

This theorem is referenced by:  rabsneu  3443