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Theorem rexsng 3673
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 3671 . 2  |-  ( A  e.  V  ->  ( E. x  e.  { A } ph  <->  [. A  /  x ]. ph ) )
2 ralsng.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32sbcieg 3023 . 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 1623    e. wcel 1684   E.wrex 2544   [.wsbc 2991   {csn 3640
This theorem is referenced by:  rexsn  3675  rexprg  3683  rextpg  3685  iunxsng  3980  frirr  4370  frsn  4760  imasng  5035  ballotlemfc0  23051  ballotlemfcc  23052  nZdef  25180  prsubrtr  25399  isconcl7a  26105  frgra2v  28177  1vwmgra  28181  elpaddat  29993  elpadd2at  29995
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-v 2790  df-sbc 2992  df-sn 3646
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