Theorem bj-elissetv 34315

Description: Version of bj-elisset 34316 with a disjoint variable condition on 𝑥, 𝑉. This proof uses only df-ex 1782, ax-gen 1797, ax-4 1811 and df-clel 2870 on top of propositional calculus. Prefer its use over bj-elisset 34316 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		dfclel 2871	. 2 ⊢ (𝐴 ∈ 𝑉 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉))
2		exsimpl 1869	. 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉) → ∃𝑥 𝑥 = 𝐴)
3	1, 2	sylbi 220	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-clel 2870

This theorem is referenced by:  bj-elisset  34316  bj-issetiv  34317  bj-ceqsaltv  34327  bj-ceqsalgv  34331  bj-spcimdvv  34336  bj-vtoclg1fv  34359  bj-vtoclg  34360  bj-ru  34379