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Theorem elissetv 2820
Description: An element of a class exists. Version of elisset 2821 with a disjoint variable condition on 𝑉, 𝑥, avoiding df-clab 2713. Prefer its use over elisset 2821 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 2815 . 2 (𝐴𝑉 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝑉))
2 exsimpl 1866 . 2 (∃𝑥(𝑥 = 𝐴𝑥𝑉) → ∃𝑥 𝑥 = 𝐴)
31, 2sylbi 217 1 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  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-ex 1777  df-clel 2814
This theorem is referenced by:  elisset  2821  clelab  2885  isseti  3496  elex  3499  elex22  3504  spcimgft  3546  bj-issetiv  36860  bj-ceqsaltv  36870  bj-ceqsalgv  36874  bj-spcimdvv  36879  bj-vtoclg1fv  36902  bj-vtoclg  36903  bj-ru  36927  bj-unexg  37021
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