Theorem rabsneu 3443

Description: Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) (Revised by Mario Carneiro, 23-Dec-2016.)

Assertion

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

Proof of Theorem rabsneu

Step	Hyp	Ref	Expression
1		df-rab 2315	. . . 4 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
2	1	eqeq1i 2047	. . 3 |- ( { x e. B | ph } = { A } <-> { x | ( x e. B /\ ph ) } = { A } )
3		absneu 3442	. . 3 |- ( ( A e. V /\ { x | ( x e. B /\ ph ) } = { A } ) -> E! x ( x e. B /\ ph ) )
4	2, 3	sylan2b 271	. 2 |- ( ( A e. V /\ { x e. B | ph } = { A } ) -> E! x ( x e. B /\ ph ) )
5		df-reu 2313	. 2 |- ( E! x e. B ph <-> E! x ( x e. B /\ ph ) )
6	4, 5	sylibr 137	1 |- ( ( A e. V /\ { x e. B | ph } = { A } ) -> E! x e. B ph )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   E!weu 1900   {cab 2026   E!wreu 2308   {crab 2310   {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-reu 2313  df-rab 2315  df-v 2559  df-sn 3381

This theorem is referenced by: (None)