Theorem bj-issetiv 32057

Description: Version of bj-isseti 32058 with a dv condition on 𝑥, 𝑉. This proof uses only df-ex 1696, ax-gen 1713, ax-4 1728 and df-clel 2606 on top of propositional calculus. Prefer its use over bj-isseti 32058 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		bj-elissetv 32055	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ ∃𝑥 𝑥 = 𝐴

Syntax hints:   = wceq 1475  ∃wex 1695   ∈ wcel 1977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-clel 2606

This theorem is referenced by:  bj-rexcom4bv  32065  bj-vtoclf  32100