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Theorem ralsn 3794
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 3791 . 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 2651   _Vcvv 2901   {csn 3759
This theorem is referenced by:  elixpsn  7039  frfi  7290  dffi3  7373  fseqenlem1  7840  fpwwe2lem13  8452  hashbc  11631  hashf1lem1  11633  rpnnen2lem11  12753  drsdirfi  14324  0subg  14894  efgsp1  15298  dprd2da  15529  lbsextlem4  16162  txkgen  17607  xkoinjcn  17642  isufil2  17863  ust0  18172  prdsxmetlem  18308  prdsbl  18413  finiunmbl  19307  xrlimcnp  20676  chtub  20865  2sqlem10  21027  dchrisum0flb  21073  pntpbnd1  21149  usgra1v  21277  constr1trl  21438  h1deoi  22901  subfacp1lem5  24651  cvmlift2lem1  24770  cvmlift2lem12  24782  heibor1lem  26211  bnj149  28586
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 2370
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 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ral 2656  df-v 2903  df-sbc 3107  df-sn 3765
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