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Theorem ralsn 3793
Description: Convert a quantification over a singleton to a substitution. (Contributed by NM, 27-Apr-2009.)
Hypotheses
Ref Expression
ralsn.1  |-  A  e. 
_V
ralsn.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ralsn  |-  ( A. x  e.  { A } ph  <->  ps )
Distinct variable groups:    x, A    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem ralsn
StepHypRef Expression
1 ralsn.1 . 2  |-  A  e. 
_V
2 ralsn.2 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32ralsng 3790 . 2  |-  ( A  e.  _V  ->  ( A. x  e.  { A } ph  <->  ps ) )
41, 3ax-mp 8 1  |-  ( A. x  e.  { A } ph  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1717   A.wral 2650   _Vcvv 2900   {csn 3758
This theorem is referenced by:  elixpsn  7038  frfi  7289  dffi3  7372  fseqenlem1  7839  fpwwe2lem13  8451  hashbc  11630  hashf1lem1  11632  rpnnen2lem11  12752  drsdirfi  14323  0subg  14893  efgsp1  15297  dprd2da  15528  lbsextlem4  16161  txkgen  17606  xkoinjcn  17641  isufil2  17862  ust0  18171  prdsxmetlem  18307  prdsbl  18412  finiunmbl  19306  xrlimcnp  20675  chtub  20864  2sqlem10  21026  dchrisum0flb  21072  pntpbnd1  21148  usgra1v  21276  constr1trl  21437  h1deoi  22900  subfacp1lem5  24650  cvmlift2lem1  24769  cvmlift2lem12  24781  heibor1lem  26210  bnj149  28585
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ral 2655  df-v 2902  df-sbc 3106  df-sn 3764
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