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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 2141 and ax-12 2177. (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 3526 1 (𝐴𝑉 → (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-clel 2816
This theorem is referenced by:  ceqsrexv  3655  clel3g  3661  elxp5  7945  xpsnen  9095  isssc  17864  metuel2  24578  isgrpo  30516  bj-finsumval0  37286  ismgmOLD  37857  brxrn  38375  pmapjat1  39855  dfatdmfcoafv2  47266
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