Theorem rabsneu 3632

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 2444	. . . 4 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
2	1	eqeq1i 2165	. . 3 |- ( { x e. B | ph } = { A } <-> { x | ( x e. B /\ ph ) } = { A } )
3		absneu 3631	. . 3 |- ( ( A e. V /\ { x | ( x e. B /\ ph ) } = { A } ) -> E! x ( x e. B /\ ph ) )
4	2, 3	sylan2b 285	. 2 |- ( ( A e. V /\ { x e. B | ph } = { A } ) -> E! x ( x e. B /\ ph ) )
5		df-reu 2442	. 2 |- ( E! x e. B ph <-> E! x ( x e. B /\ ph ) )
6	4, 5	sylibr 133	1 |- ( ( A e. V /\ { x e. B | ph } = { A } ) -> E! x e. B ph )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   E!weu 2006   e. wcel 2128   {cab 2143   E!wreu 2437   {crab 2439   {csn 3560

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-reu 2442  df-rab 2444  df-v 2714  df-sn 3566

This theorem is referenced by: (None)