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Theorem bj-elissetv 32836
Description: Version of bj-elisset 32837 with a dv condition on 𝑥, 𝑉. This proof uses only df-ex 1703, ax-gen 1720, ax-4 1735 and df-clel 2616 on top of propositional calculus. Prefer its use over bj-elisset 32837 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 2616 . 2 (𝐴𝑉 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝑉))
2 exsimpl 1793 . 2 (∃𝑥(𝑥 = 𝐴𝑥𝑉) → ∃𝑥 𝑥 = 𝐴)
31, 2sylbi 207 1 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wex 1702  wcel 1988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1703  df-clel 2616
This theorem is referenced by:  bj-elisset  32837  bj-issetiv  32838  bj-ceqsaltv  32851  bj-ceqsalgv  32855  bj-spcimdvv  32860  bj-vtoclg1fv  32887  bj-ru  32909
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