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Theorem bj-denoteslem 36814
Description: Duplicate of issettru 2815 and bj-issettruALTV 36816.

Lemma for bj-denotesALTV 36815. (Contributed by BJ, 24-Apr-2024.) (Proof modification is discouraged.)

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 2717 . . . 4 𝑥 ∈ {𝑦 ∣ ⊤}
21biantru 529 . . 3 (𝑥 = 𝐴 ↔ (𝑥 = 𝐴𝑥 ∈ {𝑦 ∣ ⊤}))
32exbii 1843 . 2 (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥(𝑥 = 𝐴𝑥 ∈ {𝑦 ∣ ⊤}))
4 dfclel 2813 . 2 (𝐴 ∈ {𝑦 ∣ ⊤} ↔ ∃𝑥(𝑥 = 𝐴𝑥 ∈ {𝑦 ∣ ⊤}))
53, 4bitr4i 278 1 (∃𝑥 𝑥 = 𝐴𝐴 ∈ {𝑦 ∣ ⊤})
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1535  wtru 1536  wex 1774  wcel 2104  {cab 2710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1538  df-ex 1775  df-sb 2061  df-clab 2711  df-clel 2812
This theorem is referenced by:  bj-denotesALTV  36815
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