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Theorem ralsn 3851
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 3848 . 2  |-  ( A  e.  _V  ->  ( A. x  e.  { A } ph  <->  ps ) )
41, 3ax-mp 5 1  |-  ( A. x  e.  { A } ph  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958   {csn 3816
This theorem is referenced by:  elixpsn  7104  frfi  7355  dffi3  7439  fseqenlem1  7910  fpwwe2lem13  8522  hashbc  11707  hashf1lem1  11709  rpnnen2lem11  12829  drsdirfi  14400  0subg  14970  efgsp1  15374  dprd2da  15605  lbsextlem4  16238  txkgen  17689  xkoinjcn  17724  isufil2  17945  ust0  18254  prdsxmetlem  18403  prdsbl  18526  finiunmbl  19443  xrlimcnp  20812  chtub  21001  2sqlem10  21163  dchrisum0flb  21209  pntpbnd1  21285  usgra1v  21414  constr1trl  21593  h1deoi  23056  subfacp1lem5  24875  cvmlift2lem1  24994  cvmlift2lem12  25006  heibor1lem  26532  cshw1  28309  cshwssizelem1  28314  bnj149  29320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-v 2960  df-sbc 3164  df-sn 3822
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