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Theorem rabtru 3551
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 nfcv 2939 . . . 4 𝑥𝑦
2 rabtru.1 . . . 4 𝑥𝐴
3 nftru 1900 . . . 4 𝑥
4 biidd 254 . . . 4 (𝑥 = 𝑦 → (⊤ ↔ ⊤))
51, 2, 3, 4elrabf 3550 . . 3 (𝑦 ∈ {𝑥𝐴 ∣ ⊤} ↔ (𝑦𝐴 ∧ ⊤))
6 tru 1658 . . . 4
76biantru 526 . . 3 (𝑦𝐴 ↔ (𝑦𝐴 ∧ ⊤))
85, 7bitr4i 270 . 2 (𝑦 ∈ {𝑥𝐴 ∣ ⊤} ↔ 𝑦𝐴)
98eqriv 2794 1 {𝑥𝐴 ∣ ⊤} = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 385   = wceq 1653  wtru 1654  wcel 2157  wnfc 2926  {crab 3091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-rab 3096  df-v 3385
This theorem is referenced by:  mptexgf  6712  aciunf1  29974
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