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Theorem rexsng 4673
Description: Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.) Avoid ax-10 2130, ax-12 2167. (Revised by GG, 30-Sep-2024.)
Hypothesis
Ref Expression
ralsng.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
rexsng (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝜑𝜓))
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem rexsng
StepHypRef Expression
1 ralsng.1 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
21notbid 317 . . 3 (𝑥 = 𝐴 → (¬ 𝜑 ↔ ¬ 𝜓))
32ralsng 4672 . 2 (𝐴𝑉 → (∀𝑥 ∈ {𝐴} ¬ 𝜑 ↔ ¬ 𝜓))
4 dfrex2 3063 . . 3 (∃𝑥 ∈ {𝐴}𝜑 ↔ ¬ ∀𝑥 ∈ {𝐴} ¬ 𝜑)
5 bicom1 220 . . . 4 ((∀𝑥 ∈ {𝐴} ¬ 𝜑 ↔ ¬ 𝜓) → (¬ 𝜓 ↔ ∀𝑥 ∈ {𝐴} ¬ 𝜑))
65con1bid 354 . . 3 ((∀𝑥 ∈ {𝐴} ¬ 𝜑 ↔ ¬ 𝜓) → (¬ ∀𝑥 ∈ {𝐴} ¬ 𝜑𝜓))
74, 6bitrid 282 . 2 ((∀𝑥 ∈ {𝐴} ¬ 𝜑 ↔ ¬ 𝜓) → (∃𝑥 ∈ {𝐴}𝜑𝜓))
83, 7syl 17 1 (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1534  wcel 2099  wral 3051  wrex 3060  {csn 4623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-v 3464  df-sn 4624
This theorem is referenced by:  rexsn  4681  rextpg  4698  iunxsng  5090  frirr  5651  frsn  5761  imasng  6085  naddunif  8715  scshwfzeqfzo  14830  dvdsprmpweqnn  16882  mnd1  18764  grp1  19037  pzriprnglem3  21469  pzriprnglem10  21476  psdmul  22156  elntg2  28916  1loopgrvd0  29438  1egrvtxdg0  29445  nfrgr2v  30202  1vwmgr  30206  elgrplsmsn  33271  grplsmid  33285  ballotlemfc0  34339  ballotlemfcc  34340  bj-restsn  36802  elrnressn  37984  elpaddat  39516  elpadd2at  39518  brfvidRP  43392  mnuunid  43988  iccelpart  47041  zlidlring  47647  lco0  47846
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