Theorem elissetv 2813

Description: An element of a class exists. Version of elisset 2814 with a disjoint variable condition on 𝑉, 𝑥, avoiding df-clab 2709. Prefer its use over elisset 2814 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 1870	. 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 395   = wceq 1540  ∃wex 1780   ∈ wcel 2105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1781  df-clel 2809

This theorem is referenced by:  elisset  2814  clelab  2878  isseti  3489  elex22  3496  bj-elissetALT  36072  bj-issetiv  36073  bj-ceqsaltv  36083  bj-ceqsalgv  36087  bj-spcimdvv  36092  bj-vtoclg1fv  36115  bj-vtoclg  36116  bj-ru  36141  bj-unexg  36235