Theorem bj-issetiv 36872

Description: Version of bj-isseti 36873 with a disjoint variable condition on 𝑥, 𝑉. The hypothesis uses 𝑉 instead of V for extra generality. This is indeed more general than isseti 3497 as long as elex 3500 is not available (and the non-dependence of bj-issetiv 36872 on special properties of the universal class V is obvious). Prefer its use over bj-isseti 36873 when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-issetiv.1	⊢ 𝐴 ∈ 𝑉

Assertion

Ref	Expression
bj-issetiv	⊢ ∃𝑥 𝑥 = 𝐴

Distinct variable groups:   𝑥,𝐴   𝑥,𝑉

Proof of Theorem bj-issetiv

Step	Hyp	Ref	Expression
1		bj-issetiv.1	. 2 ⊢ 𝐴 ∈ 𝑉
2		elissetv 2821	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ ∃𝑥 𝑥 = 𝐴

Syntax hints:   = wceq 1538  ∃wex 1777   ∈ wcel 2107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-clel 2815

This theorem is referenced by:  bj-rexcom4bv  36877  bj-vtoclf  36910