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Theorem rabtru 3689
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 1544 . . 3
2 nfcv 2905 . . . 4 𝑥𝑦
3 rabtru.1 . . . 4 𝑥𝐴
4 nftru 1804 . . . 4 𝑥
5 biidd 262 . . . 4 (𝑥 = 𝑦 → (⊤ ↔ ⊤))
62, 3, 4, 5elrabf 3688 . . 3 (𝑦 ∈ {𝑥𝐴 ∣ ⊤} ↔ (𝑦𝐴 ∧ ⊤))
71, 6mpbiran2 710 . 2 (𝑦 ∈ {𝑥𝐴 ∣ ⊤} ↔ 𝑦𝐴)
87eqriv 2734 1 {𝑥𝐴 ∣ ⊤} = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wtru 1541  wcel 2108  wnfc 2890  {crab 3436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rab 3437  df-v 3482
This theorem is referenced by:  mptexgf  7242  aciunf1  32673
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