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Theorem bj-elissetv 34184
Description: Version of bj-elisset 34185 with a disjoint variable condition on 𝑥, 𝑉. This proof uses only df-ex 1775, ax-gen 1790, ax-4 1804 and df-clel 2891 on top of propositional calculus. Prefer its use over bj-elisset 34185 when sufficient. (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-elissetv (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑉

Proof of Theorem bj-elissetv
StepHypRef Expression
1 dfclel 2892 . 2 (𝐴𝑉 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝑉))
2 exsimpl 1863 . 2 (∃𝑥(𝑥 = 𝐴𝑥𝑉) → ∃𝑥 𝑥 = 𝐴)
31, 2sylbi 219 1 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1531  wex 1774  wcel 2108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1775  df-clel 2891
This theorem is referenced by:  bj-elisset  34185  bj-issetiv  34186  bj-ceqsaltv  34196  bj-ceqsalgv  34200  bj-spcimdvv  34205  bj-vtoclg1fv  34228  bj-vtoclg  34229  bj-ru  34248
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