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Theorem rexsng 4190
Description: Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.)
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 rexsns 4188 . 2 (∃𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑)
2 ralsng.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
32sbcieg 3450 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
41, 3syl5bb 272 1 (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wcel 1987  wrex 2908  [wsbc 3417  {csn 4148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-v 3188  df-sbc 3418  df-sn 4149
This theorem is referenced by:  rexsn  4194  rexprg  4206  rextpg  4208  iunxsng  4568  frirr  5051  frsn  5150  imasng  5446  scshwfzeqfzo  13509  dvdsprmpweqnn  15513  mnd1  17252  grp1  17443  1loopgrvd0  26286  1egrvtxdg0  26293  nfrgr2v  27000  1vwmgr  27004  ballotlemfc0  30332  ballotlemfcc  30333  bj-restsn  32669  elpaddat  34567  elpadd2at  34569  brfvidRP  37458  iccelpart  40664  zlidlring  41213  lco0  41501
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