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Theorem ralsn 4582
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 4576 . 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 209   = wceq 1538  wcel 2112  wral 3109  Vcvv 3444  {csn 4528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ral 3114  df-v 3446  df-sbc 3724  df-sn 4529
This theorem is referenced by:  elixpsn  8488  frfi  8751  dffi3  8883  fseqenlem1  9439  fpwwe2lem13  10057  hashbc  13811  hashf1lem1  13813  eqs1  13961  cshw1  14179  rpnnen2lem11  15573  drsdirfi  17544  0subg  18300  efgsp1  18859  dprd2da  19161  lbsextlem4  19930  ply1coe  20929  mat0dimcrng  21079  txkgen  22261  xkoinjcn  22296  isufil2  22517  ust0  22829  prdsxmetlem  22979  prdsbl  23102  finiunmbl  24152  xrlimcnp  25558  chtub  25800  2sqlem10  26016  dchrisum0flb  26098  pntpbnd1  26174  usgr1e  27039  nbgr2vtx1edg  27144  nbuhgr2vtx1edgb  27146  wlkl1loop  27431  crctcshwlkn0lem7  27606  2pthdlem1  27720  rusgrnumwwlkl1  27758  clwwlkccatlem  27778  clwwlkn2  27833  clwwlkel  27835  clwwlkwwlksb  27843  1wlkdlem4  27929  h1deoi  29336  bnj149  32261  subfacp1lem5  32545  cvmlift2lem1  32663  cvmlift2lem12  32675  conway  33378  etasslt  33388  slerec  33391  lindsenlbs  35051  poimirlem28  35084  poimirlem32  35088  heibor1lem  35246
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