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Theorem rexsn 3675
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 3673 . 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 176    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788   {csn 3640
This theorem is referenced by:  elsnres  4991  oarec  6560  snec  6722  zornn0g  8132  fpwwe2lem13  8264  elreal  8753  vdwlem6  13033  restsn  16901  snclseqg  17798  eldm3  24119  iscst4  25177  rexsnOLD  26343  heiborlem3  26537
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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