Theorem rabsneu 4264

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 2921	. . . 4 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
2	1	eqeq1i 2627	. . 3 |- ( { x e. B | ph } = { A } <-> { x | ( x e. B /\ ph ) } = { A } )
3		absneu 4263	. . 3 |- ( ( A e. V /\ { x | ( x e. B /\ ph ) } = { A } ) -> E! x ( x e. B /\ ph ) )
4	2, 3	sylan2b 492	. 2 |- ( ( A e. V /\ { x e. B | ph } = { A } ) -> E! x ( x e. B /\ ph ) )
5		df-reu 2919	. 2 |- ( E! x e. B ph <-> E! x ( x e. B /\ ph ) )
6	4, 5	sylibr 224	1 |- ( ( A e. V /\ { x e. B | ph } = { A } ) -> E! x e. B ph )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E!weu 2470   {cab 2608   E!wreu 2914   {crab 2916   {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-reu 2919  df-rab 2921  df-v 3202  df-sn 4178

This theorem is referenced by: (None)