Theorem bj-isseti 36922

Description: Version of isseti 3454 with a class variable 𝑉 in the hypothesis instead of V for extra generality. This is indeed more general than isseti 3454 as long as elex 3457 is not available (and the non-dependence of bj-isseti 36922 on special properties of the universal class V is obvious). Use bj-issetiv 36921 instead when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-isseti.1	⊢ 𝐴 ∈ 𝑉

Assertion

Ref	Expression
bj-isseti	⊢ ∃𝑥 𝑥 = 𝐴

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem bj-isseti

Step	Hyp	Ref	Expression
1		bj-isseti.1	. 2 ⊢ 𝐴 ∈ 𝑉
2		elisset 2813	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ ∃𝑥 𝑥 = 𝐴

Syntax hints:   = wceq 1541  ∃wex 1780   ∈ wcel 2111

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 2113

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-clel 2806

This theorem is referenced by:  bj-rexcom4b  36927