Theorem bj-elissetv 34186

Description: Version of bj-elisset 34187 with a disjoint variable condition on 𝑥, 𝑉. This proof uses only df-ex 1777, ax-gen 1792, ax-4 1806 and df-clel 2893 on top of propositional calculus. Prefer its use over bj-elisset 34187 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 2894	. 2 ⊢ (𝐴 ∈ 𝑉 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉))
2		exsimpl 1865	. 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉) → ∃𝑥 𝑥 = 𝐴)
3	1, 2	sylbi 219	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-clel 2893

This theorem is referenced by:  bj-elisset  34187  bj-issetiv  34188  bj-ceqsaltv  34198  bj-ceqsalgv  34202  bj-spcimdvv  34207  bj-vtoclg1fv  34230  bj-vtoclg  34231  bj-ru  34250