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Theorem bj-denoteslem 34792
Description: Lemma for bj-denotes 34793. (Contributed by BJ, 24-Apr-2024.)
Assertion
Ref Expression
bj-denoteslem (∃𝑥 𝑥 = 𝐴𝐴 ∈ {𝑦 ∣ ⊤})
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem bj-denoteslem
StepHypRef Expression
1 vextru 2721 . . . 4 𝑥 ∈ {𝑦 ∣ ⊤}
21biantru 533 . . 3 (𝑥 = 𝐴 ↔ (𝑥 = 𝐴𝑥 ∈ {𝑦 ∣ ⊤}))
32exbii 1855 . 2 (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥(𝑥 = 𝐴𝑥 ∈ {𝑦 ∣ ⊤}))
4 dfclel 2817 . 2 (𝐴 ∈ {𝑦 ∣ ⊤} ↔ ∃𝑥(𝑥 = 𝐴𝑥 ∈ {𝑦 ∣ ⊤}))
53, 4bitr4i 281 1 (∃𝑥 𝑥 = 𝐴𝐴 ∈ {𝑦 ∣ ⊤})
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1543  wtru 1544  wex 1787  wcel 2110  {cab 2714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-clel 2816
This theorem is referenced by:  bj-denotes  34793  bj-issettru  34794  bj-elabtru  34795
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