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Theorem ralsn 3674
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 3672 . 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 176    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   {csn 3640
This theorem is referenced by:  elixpsn  6855  frfi  7102  dffi3  7184  fseqenlem1  7651  fpwwe2lem13  8264  hashbc  11391  hashf1lem1  11393  rpnnen2lem11  12503  drsdirfi  14072  0subg  14642  efgsp1  15046  dprd2da  15277  lbsextlem4  15914  txkgen  17346  xkoinjcn  17381  isufil2  17603  prdsxmetlem  17932  prdsbl  18037  finiunmbl  18901  xrlimcnp  20263  chtub  20451  2sqlem10  20613  dchrisum0flb  20659  pntpbnd1  20735  h1deoi  22128  subfacp1lem5  23715  cvmlift2lem1  23833  cvmlift2lem12  23845  basexre  25522  heibor1lem  26533  usgra1v  28126  bnj149  28907
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-ral 2548  df-v 2790  df-sbc 2992  df-sn 3646
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