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Theorem rabtru 28723
 Description: Abtract builder using the constant wff ⊤ (Contributed by Thierry Arnoux, 4-May-2020.)
Hypothesis
Ref Expression
rabtru.1 𝑥𝐴
Assertion
Ref Expression
rabtru {𝑥𝐴 ∣ ⊤} = 𝐴

Proof of Theorem rabtru
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nfcv 2751 . . . 4 𝑥𝑦
2 rabtru.1 . . . 4 𝑥𝐴
3 nftru 1721 . . . 4 𝑥
4 biidd 251 . . . 4 (𝑥 = 𝑦 → (⊤ ↔ ⊤))
51, 2, 3, 4elrabf 3329 . . 3 (𝑦 ∈ {𝑥𝐴 ∣ ⊤} ↔ (𝑦𝐴 ∧ ⊤))
6 tru 1479 . . . 4
76biantru 525 . . 3 (𝑦𝐴 ↔ (𝑦𝐴 ∧ ⊤))
85, 7bitr4i 266 . 2 (𝑦 ∈ {𝑥𝐴 ∣ ⊤} ↔ 𝑦𝐴)
98eqriv 2607 1 {𝑥𝐴 ∣ ⊤} = 𝐴
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977  Ⅎwnfc 2738  {crab 2900 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175 This theorem is referenced by:  mptexgf  28809  aciunf1  28845
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