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Theorem rexsn 3818
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 3815 . 2  |-  ( A  e.  _V  ->  ( E. x  e.  { A } ph  <->  ps ) )
41, 3ax-mp 8 1  |-  ( E. x  e.  { A } ph  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1721   E.wrex 2675   _Vcvv 2924   {csn 3782
This theorem is referenced by:  elsnres  5149  oarec  6772  snec  6934  zornn0g  8349  fpwwe2lem13  8481  elreal  8970  vdwlem6  13317  restsn  17196  snclseqg  18106  ust0  18210  eldm3  25341  heiborlem3  26420
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-rex 2680  df-v 2926  df-sbc 3130  df-sn 3788
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