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Theorem bj-elissetv 32055
 Description: Version of bj-elisset 32056 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-elisset 32056 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 df-clel 2606 . 2 (𝐴𝑉 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝑉))
2 exsimpl 1783 . 2 (∃𝑥(𝑥 = 𝐴𝑥𝑉) → ∃𝑥 𝑥 = 𝐴)
31, 2sylbi 206 1 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = 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-elisset  32056  bj-issetiv  32057  bj-ceqsaltv  32070  bj-ceqsalgv  32074  bj-vtoclg1fv  32104  bj-ru  32126
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