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Theorem ralsng 4612
Description: Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.) (Proof shortened by AV, 7-Apr-2023.)
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 nfv 1911 . 2 𝑥𝜓
2 ralsng.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
31, 2ralsngf 4610 1 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208   = wceq 1533  wcel 2110  wral 3138  {csn 4566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-v 3496  df-sbc 3772  df-sn 4567
This theorem is referenced by:  2ralsng  4615  ralsn  4618  raltpg  4633  ralunsn  4823  iinxsng  5009  frirr  5531  posn  5636  frsn  5638  f12dfv  7029  ranksnb  9255  mgm1  17867  sgrp1  17909  mnd1  17951  grp1  18205  cntzsnval  18453  abl1  18985  srgbinomlem4  19292  ring1  19351  mat1dimmul  21084  ufileu  22526  istrkg3ld  26246  1hevtxdg0  27286  wlkp1lem8  27461  wwlksnext  27670  wwlksext2clwwlk  27835  dfconngr1  27966  1conngr  27972  frgr1v  28049  lindssn  30939  lbslsat  31014  bj-raldifsn  34391  lindsadd  34884  poimirlem26  34917  poimirlem27  34918  poimirlem31  34922  zlidlring  44200  linds0  44521  snlindsntor  44527  lmod1  44548
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