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Theorem ralsng 4680
Description: Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.) Avoid ax-10 2139, ax-12 2175. (Revised by GG, 30-Sep-2024.)
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 df-ral 3060 . . 3 (∀𝑥 ∈ {𝐴}𝜑 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝜑))
2 velsn 4647 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
32imbi1i 349 . . . 4 ((𝑥 ∈ {𝐴} → 𝜑) ↔ (𝑥 = 𝐴𝜑))
43albii 1816 . . 3 (∀𝑥(𝑥 ∈ {𝐴} → 𝜑) ↔ ∀𝑥(𝑥 = 𝐴𝜑))
51, 4bitri 275 . 2 (∀𝑥 ∈ {𝐴}𝜑 ↔ ∀𝑥(𝑥 = 𝐴𝜑))
6 elisset 2821 . . 3 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
7 ralsng.1 . . . . . . 7 (𝑥 = 𝐴 → (𝜑𝜓))
87pm5.74i 271 . . . . . 6 ((𝑥 = 𝐴𝜑) ↔ (𝑥 = 𝐴𝜓))
98albii 1816 . . . . 5 (∀𝑥(𝑥 = 𝐴𝜑) ↔ ∀𝑥(𝑥 = 𝐴𝜓))
109a1i 11 . . . 4 (∃𝑥 𝑥 = 𝐴 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ ∀𝑥(𝑥 = 𝐴𝜓)))
11 19.23v 1940 . . . . 5 (∀𝑥(𝑥 = 𝐴𝜓) ↔ (∃𝑥 𝑥 = 𝐴𝜓))
1211a1i 11 . . . 4 (∃𝑥 𝑥 = 𝐴 → (∀𝑥(𝑥 = 𝐴𝜓) ↔ (∃𝑥 𝑥 = 𝐴𝜓)))
13 pm5.5 361 . . . 4 (∃𝑥 𝑥 = 𝐴 → ((∃𝑥 𝑥 = 𝐴𝜓) ↔ 𝜓))
1410, 12, 133bitrd 305 . . 3 (∃𝑥 𝑥 = 𝐴 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
156, 14syl 17 . 2 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
165, 15bitrid 283 1 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535   = wceq 1537  wex 1776  wcel 2106  wral 3059  {csn 4631
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-v 3480  df-sn 4632
This theorem is referenced by:  rexsng  4681  2ralsng  4683  ralsn  4686  ralprg  4701  raltpg  4703  ralunsn  4899  iinxsng  5093  frirr  5665  posn  5774  frsn  5776  f1ounsn  7292  f12dfv  7293  naddov2  8716  naddunif  8730  naddasslem1  8731  naddasslem2  8732  ranksnb  9865  mgm1  18684  sgrp1  18755  mnd1  18805  grp1  19078  cntzsnval  19355  abl1  19899  srgbinomlem4  20247  ring1  20324  mat1dimmul  22498  ufileu  23943  istrkg3ld  28484  1hevtxdg0  29538  wlkp1lem8  29713  wwlksnext  29923  wwlksext2clwwlk  30086  dfconngr1  30217  1conngr  30223  frgr1v  30300  lindssn  33386  lbslsat  33644  bj-raldifsn  37083  lindsadd  37600  poimirlem26  37633  poimirlem27  37634  poimirlem31  37638  cantnfresb  43314  safesnsupfilb  43408  cfsetsnfsetf1  47009  zlidlring  48078  linds0  48311  snlindsntor  48317  lmod1  48338
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