Theorem bj-elissetv 32861

Description: Version of bj-elisset 32862 with a dv condition on 𝑥, 𝑉. This proof uses only df-ex 1705, ax-gen 1722, ax-4 1737 and df-clel 2618 on top of propositional calculus. Prefer its use over bj-elisset 32862 when sufficient. (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-elissetv	⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑉

Proof of Theorem bj-elissetv

Step	Hyp	Ref	Expression
1		df-clel 2618	. 2 ⊢ (𝐴 ∈ 𝑉 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉))
2		exsimpl 1795	. 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉) → ∃𝑥 𝑥 = 𝐴)
3	1, 2	sylbi 207	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-clel 2618

This theorem is referenced by:  bj-elisset  32862  bj-issetiv  32863  bj-ceqsaltv  32876  bj-ceqsalgv  32880  bj-spcimdvv  32885  bj-vtoclg1fv  32912  bj-ru  32934