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Theorem ralsn 4611
Description: Convert a quantification over 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 4605 . 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 207   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  {csn 4557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-v 3494  df-sbc 3770  df-sn 4558
This theorem is referenced by:  elixpsn  8489  frfi  8751  dffi3  8883  fseqenlem1  9438  fpwwe2lem13  10052  hashbc  13799  hashf1lem1  13801  eqs1  13954  cshw1  14172  rpnnen2lem11  15565  drsdirfi  17536  0subg  18242  efgsp1  18792  dprd2da  19093  lbsextlem4  19862  ply1coe  20392  mat0dimcrng  21007  txkgen  22188  xkoinjcn  22223  isufil2  22444  ust0  22755  prdsxmetlem  22905  prdsbl  23028  finiunmbl  24072  xrlimcnp  25473  chtub  25715  2sqlem10  25931  dchrisum0flb  26013  pntpbnd1  26089  usgr1e  26954  nbgr2vtx1edg  27059  nbuhgr2vtx1edgb  27061  wlkl1loop  27346  crctcshwlkn0lem7  27521  2pthdlem1  27636  rusgrnumwwlkl1  27674  clwwlkccatlem  27694  clwwlkn2  27749  clwwlkel  27752  clwwlkwwlksb  27760  1wlkdlem4  27846  h1deoi  29253  bnj149  32046  subfacp1lem5  32328  cvmlift2lem1  32446  cvmlift2lem12  32458  conway  33161  etasslt  33171  slerec  33174  lindsenlbs  34768  poimirlem28  34801  poimirlem32  34805  heibor1lem  34968
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