Theorem elissetv 2844

Description: An element of a class exists. Version of elisset 2845 with a disjoint variable condition on 𝑉, 𝑥, avoiding df-clab 2742. Prefer its use over elisset 2845 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 2839	. 2 ⊢ (𝐴 ∈ 𝑉 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉))
2		exsimpl 1889	. 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 399   = wceq 1561  ∃wex 1800   ∈ wcel 2143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1801  df-clel 2838

This theorem is referenced by:  elisset  2845  clelab  2907  isseti  3473  elex  3476  elex22  3479  spcimgft  3515  bj-issetiv  37367  bj-ceqsaltv  37377  bj-ceqsalgv  37381  bj-spcimdvv  37386  bj-vtoclg1fv  37409  bj-vtoclg  37410  bj-ru  37434  bj-unexg  37528