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Theorem ralsng 3564
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 3562 . 2 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
2 ralsng.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
32sbcieg 2941 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
41, 3bitrd 187 1 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1331  wcel 1480  wral 2416  [wsbc 2909  {csn 3527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-sbc 2910  df-sn 3533
This theorem is referenced by:  ralsn  3567  ralprg  3574  raltpg  3576  ralunsn  3724  iinxsng  3886  posng  4611  fimax2gtrilemstep  6794  iseqf1olemqk  10274  seq3f1olemstep  10281  fimaxre2  11005  nninfsellemdc  13236
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