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Theorem rexsng 3686
Description: Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.)
Hypothesis
Ref Expression
ralsng.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
rexsng  |-  ( A  e.  V  ->  ( E. x  e.  { A } ph  <->  ps ) )
Distinct variable groups:    x, A    ps, x
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem rexsng
StepHypRef Expression
1 rexsns 3684 . 2  |-  ( A  e.  V  ->  ( E. x  e.  { A } ph  <->  [. A  /  x ]. ph ) )
2 ralsng.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32sbcieg 3036 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ps ) )
41, 3bitrd 244 1  |-  ( A  e.  V  ->  ( E. x  e.  { A } ph  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   E.wrex 2557   [.wsbc 3004   {csn 3653
This theorem is referenced by:  rexsn  3688  rexprg  3696  rextpg  3698  iunxsng  3996  frirr  4386  frsn  4776  imasng  5051  ballotlemfc0  23067  ballotlemfcc  23068  nZdef  25283  prsubrtr  25502  isconcl7a  26208  frgra2v  28423  1vwmgra  28427  elpaddat  30615  elpadd2at  30617
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-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-v 2803  df-sbc 3005  df-sn 3659
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