Theorem elissetv 2821

Description: An element of a class exists. Version of elisset 2822 with a disjoint variable condition on 𝑉, 𝑥, avoiding df-clab 2718. Prefer its use over elisset 2822 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 2819	. 2 ⊢ (𝐴 ∈ 𝑉 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉))
2		exsimpl 1875	. 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉) → ∃𝑥 𝑥 = 𝐴)
3	1, 2	sylbi 216	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1542  ∃wex 1786   ∈ wcel 2110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1787  df-clel 2818

This theorem is referenced by:  elisset  2822  isseti  3446  bj-elissetALT  35049  bj-issetiv  35050  bj-ceqsaltv  35060  bj-ceqsalgv  35064  bj-spcimdvv  35069  bj-vtoclg1fv  35092  bj-vtoclg  35093  bj-ru  35121