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Theorem ceqsexgv 3654
Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996.) Drop ax-10 2139 and ax-12 2175. (Revised by GG, 1-Dec-2023.)
Hypothesis
Ref Expression
ceqsexgv.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ceqsexgv (𝐴𝑉 → (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem ceqsexgv
StepHypRef Expression
1 id 22 . 2 (𝑥 = 𝐴𝑥 = 𝐴)
2 ceqsexgv.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
31, 2cgsexg 3524 1 (𝐴𝑉 → (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106
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
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-clel 2814
This theorem is referenced by:  ceqsrexv  3655  clel3g  3661  elxp5  7946  xpsnen  9094  isssc  17868  metuel2  24594  isgrpo  30526  bj-finsumval0  37268  ismgmOLD  37837  brxrn  38356  pmapjat1  39836  dfatdmfcoafv2  47204
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