Theorem elissetv 2833

Description: An element of a class exists. Version of elisset 2834 with a disjoint variable condition on 𝑉, 𝑥, avoiding df-clab 2731. Prefer its use over elisset 2834 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 2828	. 2 ⊢ (𝐴 ∈ 𝑉 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉))
2		exsimpl 1878	. 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉) → ∃𝑥 𝑥 = 𝐴)
3	1, 2	sylbi 219	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1550  ∃wex 1789   ∈ wcel 2132

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1790  df-clel 2827

This theorem is referenced by:  elisset  2834  clelab  2896  isseti  3462  elex  3465  elex22  3468  spcimgft  3504  bj-issetiv  37300  bj-ceqsaltv  37310  bj-ceqsalgv  37314  bj-spcimdvv  37319  bj-vtoclg1fv  37342  bj-vtoclg  37343  bj-ru  37367  bj-unexg  37461