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Theorem bj-denoteslem 34328
 Description: Lemma for bj-denotes 34329. (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 2783 . . . 4 𝑥 ∈ {𝑦 ∣ ⊤}
21biantru 533 . . 3 (𝑥 = 𝐴 ↔ (𝑥 = 𝐴𝑥 ∈ {𝑦 ∣ ⊤}))
32exbii 1849 . 2 (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥(𝑥 = 𝐴𝑥 ∈ {𝑦 ∣ ⊤}))
4 dfclel 2871 . 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 1538  ⊤wtru 1539  ∃wex 1781   ∈ wcel 2111  {cab 2776 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-clel 2870 This theorem is referenced by:  bj-denotes  34329  bj-issettru  34330  bj-elabtru  34331
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