Theorem elissetv 2812

Description: An element of a class exists. Version of elisset 2813 with a disjoint variable condition on 𝑉, 𝑥, avoiding df-clab 2708. Prefer its use over elisset 2813 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 2809	. 2 ⊢ (𝐴 ∈ 𝑉 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉))
2		exsimpl 1869	. 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 394   = wceq 1539  ∃wex 1779   ∈ wcel 2104

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 1911  ax-6 1969  ax-7 2009  ax-8 2106

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1780  df-clel 2808

This theorem is referenced by:  elisset  2813  clelab  2877  isseti  3488  elex22  3495  bj-elissetALT  36059  bj-issetiv  36060  bj-ceqsaltv  36070  bj-ceqsalgv  36074  bj-spcimdvv  36079  bj-vtoclg1fv  36102  bj-vtoclg  36103  bj-ru  36128  bj-unexg  36222