Theorem elissetv 2818

Description: An element of a class exists. Version of elisset 2819 with a disjoint variable condition on 𝑉, 𝑥, avoiding df-clab 2716. Prefer its use over elisset 2819 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 2813	. 2 ⊢ (𝐴 ∈ 𝑉 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉))
2		exsimpl 1870	. 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 1542  ∃wex 1781   ∈ wcel 2114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-clel 2812

This theorem is referenced by:  elisset  2819  clelab  2881  isseti  3459  elex  3462  elex22  3466  spcimgft  3504  bj-issetiv  37053  bj-ceqsaltv  37063  bj-ceqsalgv  37067  bj-spcimdvv  37072  bj-vtoclg1fv  37095  bj-vtoclg  37096  bj-ru  37120  bj-unexg  37214