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Theorem rexsn 4067
Description: Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.)
Hypotheses
Ref Expression
ralsn.1  |-  A  e. 
_V
ralsn.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
rexsn  |-  ( E. x  e.  { A } ph  <->  ps )
Distinct variable groups:    x, A    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem rexsn
StepHypRef Expression
1 ralsn.1 . 2  |-  A  e. 
_V
2 ralsn.2 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32rexsng 4063 . 2  |-  ( A  e.  _V  ->  ( E. x  e.  { A } ph  <->  ps ) )
41, 3ax-mp 5 1  |-  ( E. x  e.  { A } ph  <->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767   E.wrex 2815   _Vcvv 3113   {csn 4027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-rex 2820  df-v 3115  df-sbc 3332  df-sn 4028
This theorem is referenced by:  elsnres  5310  oarec  7211  snec  7374  zornn0g  8885  fpwwe2lem13  9020  elreal  9508  vdwlem6  14363  pmatcollpw3fi1  19084  restsn  19465  snclseqg  20377  ust0  20485  eulerpartlemgh  27985  eldm3  28796  heiborlem3  29940
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