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Theorem elissetv 2815
Description: An element of a class exists. Version of elisset 2816 with a disjoint variable condition on 𝑉, 𝑥, avoiding df-clab 2711. Prefer its use over elisset 2816 when sufficient (for instance in usages where 𝑥 is a dummy variable). (Contributed by BJ, 14-Sep-2019.)
Assertion
Ref Expression
elissetv (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑉

Proof of Theorem elissetv
StepHypRef Expression
1 dfclel 2812 . 2 (𝐴𝑉 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝑉))
2 exsimpl 1872 . 2 (∃𝑥(𝑥 = 𝐴𝑥𝑉) → ∃𝑥 𝑥 = 𝐴)
31, 2sylbi 216 1 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wex 1782  wcel 2107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-clel 2811
This theorem is referenced by:  elisset  2816  clelab  2880  isseti  3490  elex22  3497  vtoclgft  3541  bj-elissetALT  35756  bj-issetiv  35757  bj-ceqsaltv  35767  bj-ceqsalgv  35771  bj-spcimdvv  35776  bj-vtoclg1fv  35799  bj-vtoclg  35800  bj-ru  35825  bj-unexg  35919
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