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Theorem issetlem 2819
Description: Lemma for elisset 2821 and isset 3492. (Contributed by NM, 26-May-1993.) Extract from the proof of isset 3492. (Revised by WL, 2-Feb-2025.)
Hypothesis
Ref Expression
issetlem.1 𝑥𝑉
Assertion
Ref Expression
issetlem (𝐴𝑉 ↔ ∃𝑥 𝑥 = 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑉

Proof of Theorem issetlem
StepHypRef Expression
1 dfclel 2815 . 2 (𝐴𝑉 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝑉))
2 issetlem.1 . . . 4 𝑥𝑉
32biantru 529 . . 3 (𝑥 = 𝐴 ↔ (𝑥 = 𝐴𝑥𝑉))
43exbii 1845 . 2 (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝑉))
51, 4bitr4i 278 1 (𝐴𝑉 ↔ ∃𝑥 𝑥 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106
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 1908  ax-6 1965  ax-7 2005  ax-8 2108
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-clel 2814
This theorem is referenced by:  isset  3492
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