Theorem rabsneu 3665

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 2464	. . . 4 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
2	1	eqeq1i 2185	. . 3 |- ( { x e. B | ph } = { A } <-> { x | ( x e. B /\ ph ) } = { A } )
3		absneu 3664	. . 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 2462	. 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 1353   E!weu 2026   e. wcel 2148   {cab 2163   E!wreu 2457   {crab 2459   {csn 3592

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-reu 2462  df-rab 2464  df-v 2739  df-sn 3598

This theorem is referenced by: (None)