Theorem elissetv 2810

Description: An element of a class exists. Version of elisset 2811 with a disjoint variable condition on 𝑉, 𝑥, avoiding df-clab 2709. Prefer its use over elisset 2811 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 2805	. 2 ⊢ (𝐴 ∈ 𝑉 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉))
2		exsimpl 1869	. 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 1541  ∃wex 1780   ∈ wcel 2110

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 1911  ax-6 1968  ax-7 2009  ax-8 2112

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-clel 2804

This theorem is referenced by:  elisset  2811  clelab  2874  isseti  3452  elex  3455  elex22  3459  spcimgft  3499  bj-issetiv  36890  bj-ceqsaltv  36900  bj-ceqsalgv  36904  bj-spcimdvv  36909  bj-vtoclg1fv  36932  bj-vtoclg  36933  bj-ru  36957  bj-unexg  37051