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Theorem rabsneu 3879
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
StepHypRef Expression
1 df-rab 2714 . . . 4  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
21eqeq1i 2443 . . 3  |-  ( { x  e.  B  |  ph }  =  { A } 
<->  { x  |  ( x  e.  B  /\  ph ) }  =  { A } )
3 absneu 3878 . . 3  |-  ( ( A  e.  V  /\  { x  |  ( x  e.  B  /\  ph ) }  =  { A } )  ->  E! x ( x  e.  B  /\  ph )
)
42, 3sylan2b 462 . 2  |-  ( ( A  e.  V  /\  { x  e.  B  |  ph }  =  { A } )  ->  E! x ( x  e.  B  /\  ph )
)
5 df-reu 2712 . 2  |-  ( E! x  e.  B  ph  <->  E! x ( x  e.  B  /\  ph )
)
64, 5sylibr 204 1  |-  ( ( A  e.  V  /\  { x  e.  B  |  ph }  =  { A } )  ->  E! x  e.  B  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   E!weu 2281   {cab 2422   E!wreu 2707   {crab 2709   {csn 3814
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-reu 2712  df-rab 2714  df-v 2958  df-sn 3820
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