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Theorem ralsng 3478
Description: Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
Hypothesis
Ref Expression
ralsng.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ralsng (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑𝜓))
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem ralsng
StepHypRef Expression
1 ralsns 3476 . 2 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
2 ralsng.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
32sbcieg 2869 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
41, 3bitrd 186 1 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1289  wcel 1438  wral 2359  [wsbc 2838  {csn 3441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-sbc 2839  df-sn 3447
This theorem is referenced by:  ralsn  3481  ralprg  3488  raltpg  3490  ralunsn  3636  iinxsng  3798  posng  4498  fimax2gtrilemstep  6596  iseqf1olemqk  9888  seq3f1olemstep  9895  fimaxre2  10622  nninfsellemdc  11559
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