Theorem rabsneu 3739

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 2517	. . . 4 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
2	1	eqeq1i 2237	. . 3 |- ( { x e. B | ph } = { A } <-> { x | ( x e. B /\ ph ) } = { A } )
3		absneu 3738	. . 3 |- ( ( A e. V /\ { x | ( x e. B /\ ph ) } = { A } ) -> E! x ( x e. B /\ ph ) )
4	2, 3	sylan2b 287	. 2 |- ( ( A e. V /\ { x e. B | ph } = { A } ) -> E! x ( x e. B /\ ph ) )
5		df-reu 2515	. 2 |- ( E! x e. B ph <-> E! x ( x e. B /\ ph ) )
6	4, 5	sylibr 134	1 |- ( ( A e. V /\ { x e. B | ph } = { A } ) -> E! x e. B ph )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   E!weu 2077   e. wcel 2200   {cab 2215   E!wreu 2510   {crab 2512   {csn 3666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-reu 2515  df-rab 2517  df-v 2801  df-sn 3672

This theorem is referenced by: (None)