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Theorem rabsneu 3715
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 2565 . . . 4  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
21eqeq1i 2303 . . 3  |-  ( { x  e.  B  |  ph }  =  { A } 
<->  { x  |  ( x  e.  B  /\  ph ) }  =  { A } )
3 absneu 3714 . . 3  |-  ( ( A  e.  V  /\  { x  |  ( x  e.  B  /\  ph ) }  =  { A } )  ->  E! x ( x  e.  B  /\  ph )
)
42, 3sylan2b 461 . 2  |-  ( ( A  e.  V  /\  { x  e.  B  |  ph }  =  { A } )  ->  E! x ( x  e.  B  /\  ph )
)
5 df-reu 2563 . 2  |-  ( E! x  e.  B  ph  <->  E! x ( x  e.  B  /\  ph )
)
64, 5sylibr 203 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 358    = wceq 1632    e. wcel 1696   E!weu 2156   {cab 2282   E!wreu 2558   {crab 2560   {csn 3653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-reu 2563  df-rab 2565  df-v 2803  df-sn 3659
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