Theorem bj-isseti 37572

Description: Version of isseti 3475 with a class variable 𝑉 in the hypothesis instead of V for extra generality. This is indeed more general than isseti 3475 as long as elex 3478 is not available (and the non-dependence of bj-isseti 37572 on special properties of the universal class V is obvious). Use bj-issetiv 37571 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 2847	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ ∃𝑥 𝑥 = 𝐴

This proof depends on syntax axioms:   = wceq 1570  ∃wex 1812   ∈ wcel 2146

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-clel 2840

This theorem is used by:  bj-rexcom4b  37577