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 2714. 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	⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑉

Proof of Theorem elissetv

Step	Hyp	Ref	Expression
1		dfclel 2810	. 2 ⊢ (𝐴 ∈ 𝑉 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉))
2		exsimpl 1868	. 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉) → ∃𝑥 𝑥 = 𝐴)
3	1, 2	sylbi 217	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)

Syntax hints:   → wi 4   ∧ 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-ex 1780  df-clel 2809

This theorem is referenced by:  elisset  2816  clelab  2880  isseti  3477  elex  3480  elex22  3485  spcimgft  3525  bj-issetiv  36895  bj-ceqsaltv  36905  bj-ceqsalgv  36909  bj-spcimdvv  36914  bj-vtoclg1fv  36937  bj-vtoclg  36938  bj-ru  36962  bj-unexg  37056