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Theorem rexsnOLD 26446
Description: Restricted existential quantification over a singleton. (Moved to rexsn 3688 in main set.mm and may be deleted by mathbox owner, JM. --NM 29-Jan-2012.) (Contributed by Jeff Madsen, 5-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
rexsnOLD.1  |-  A  e. 
_V
rexsnOLD.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
rexsnOLD  |-  ( E. x  e.  { A } ph  <->  ps )
Distinct variable groups:    x, A    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem rexsnOLD
StepHypRef Expression
1 rexsnOLD.1 . 2  |-  A  e. 
_V
2 rexsnOLD.2 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
31, 2rexsn 3688 1  |-  ( 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   _Vcvv 2801   {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-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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