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Theorem ralsn 4629
Description: Convert a universal quantification restricted to a singleton to a substitution. (Contributed by NM, 27-Apr-2009.)
Hypotheses
Ref Expression
ralsn.1 𝐴 ∈ V
ralsn.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ralsn (∀𝑥 ∈ {𝐴}𝜑𝜓)
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ralsn
StepHypRef Expression
1 ralsn.1 . 2 𝐴 ∈ V
2 ralsn.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
32ralsng 4621 . 2 (𝐴 ∈ V → (∀𝑥 ∈ {𝐴}𝜑𝜓))
41, 3ax-mp 5 1 (∀𝑥 ∈ {𝐴}𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1540  wcel 2105  wral 3061  Vcvv 3441  {csn 4573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-v 3443  df-sn 4574
This theorem is referenced by:  elixpsn  8796  frfi  9153  dffi3  9288  ssttrcl  9572  ttrclss  9577  ttrclselem2  9583  fseqenlem1  9881  fpwwe2lem12  10499  hashbc  14265  hashf1lem1  14268  hashf1lem1OLD  14269  eqs1  14416  cshw1  14633  rpnnen2lem11  16032  drsdirfi  18120  0subg  18876  efgsp1  19438  dprd2da  19740  lbsextlem4  20529  ply1coe  21573  mat0dimcrng  21725  txkgen  22909  xkoinjcn  22944  isufil2  23165  ust0  23477  prdsxmetlem  23627  prdsbl  23753  finiunmbl  24814  xrlimcnp  26224  chtub  26466  2sqlem10  26682  dchrisum0flb  26764  pntpbnd1  26840  conway  27044  etasslt  27058  slerec  27064  bday1s  27076  usgr1e  27901  nbgr2vtx1edg  28006  nbuhgr2vtx1edgb  28008  wlkl1loop  28294  crctcshwlkn0lem7  28469  2pthdlem1  28583  rusgrnumwwlkl1  28621  clwwlkccatlem  28641  clwwlkn2  28696  clwwlkel  28698  clwwlkwwlksb  28706  1wlkdlem4  28792  h1deoi  30199  bnj149  33154  subfacp1lem5  33445  cvmlift2lem1  33563  cvmlift2lem12  33575  xpord2ind  34076  xpord3ind  34082  naddcllem  34110  madebdaylemlrcut  34175  lindsenlbs  35877  poimirlem28  35910  poimirlem32  35914  heibor1lem  36072
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