Theorem bj-issetiv 37300

Description: Version of bj-isseti 37301 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 3462 as long as elex 3465 is not available (and the non-dependence of bj-issetiv 37300 on special properties of the universal class V is obvious). Prefer its use over bj-isseti 37301 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 2833	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ ∃𝑥 𝑥 = 𝐴

Syntax hints:   = 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:  bj-rexcom4bv  37305  bj-vtoclf  37338