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Theorem rabsneu 3510
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 2368 . . . 4  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
21eqeq1i 2095 . . 3  |-  ( { x  e.  B  |  ph }  =  { A } 
<->  { x  |  ( x  e.  B  /\  ph ) }  =  { A } )
3 absneu 3509 . . 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 2366 . 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 1289    e. wcel 1438   E!weu 1948   {cab 2074   E!wreu 2361   {crab 2363   {csn 3441
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-reu 2366  df-rab 2368  df-v 2621  df-sn 3447
This theorem is referenced by: (None)
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