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Theorem rabsneu 3492
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 2364 . . . 4  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
21eqeq1i 2092 . . 3  |-  ( { x  e.  B  |  ph }  =  { A } 
<->  { x  |  ( x  e.  B  /\  ph ) }  =  { A } )
3 absneu 3491 . . 3  |-  ( ( A  e.  V  /\  { x  |  ( x  e.  B  /\  ph ) }  =  { A } )  ->  E! x ( x  e.  B  /\  ph )
)
42, 3sylan2b 281 . 2  |-  ( ( A  e.  V  /\  { x  e.  B  |  ph }  =  { A } )  ->  E! x ( x  e.  B  /\  ph )
)
5 df-reu 2362 . 2  |-  ( E! x  e.  B  ph  <->  E! x ( x  e.  B  /\  ph )
)
64, 5sylibr 132 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 102    = wceq 1287    e. wcel 1436   E!weu 1945   {cab 2071   E!wreu 2357   {crab 2359   {csn 3425
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-reu 2362  df-rab 2364  df-v 2616  df-sn 3431
This theorem is referenced by: (None)
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