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Theorem rabtru 3621
Description: Abstract 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 tru 1543 . . 3
2 nfcv 2907 . . . 4 𝑥𝑦
3 rabtru.1 . . . 4 𝑥𝐴
4 nftru 1807 . . . 4 𝑥
5 biidd 261 . . . 4 (𝑥 = 𝑦 → (⊤ ↔ ⊤))
62, 3, 4, 5elrabf 3620 . . 3 (𝑦 ∈ {𝑥𝐴 ∣ ⊤} ↔ (𝑦𝐴 ∧ ⊤))
71, 6mpbiran2 707 . 2 (𝑦 ∈ {𝑥𝐴 ∣ ⊤} ↔ 𝑦𝐴)
87eqriv 2735 1 {𝑥𝐴 ∣ ⊤} = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  wtru 1540  wcel 2106  wnfc 2887  {crab 3068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-rab 3073  df-v 3434
This theorem is referenced by:  mptexgf  7098  aciunf1  31000
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