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Theorem rabtru 3466
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 2866 . . . 4 𝑥𝑦
2 rabtru.1 . . . 4 𝑥𝐴
3 nftru 1843 . . . 4 𝑥
4 biidd 252 . . . 4 (𝑥 = 𝑦 → (⊤ ↔ ⊤))
51, 2, 3, 4elrabf 3465 . . 3 (𝑦 ∈ {𝑥𝐴 ∣ ⊤} ↔ (𝑦𝐴 ∧ ⊤))
6 tru 1600 . . . 4
76biantru 527 . . 3 (𝑦𝐴 ↔ (𝑦𝐴 ∧ ⊤))
85, 7bitr4i 267 . 2 (𝑦 ∈ {𝑥𝐴 ∣ ⊤} ↔ 𝑦𝐴)
98eqriv 2721 1 {𝑥𝐴 ∣ ⊤} = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1596  wtru 1597  wcel 2103  wnfc 2853  {crab 3018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-rab 3023  df-v 3306
This theorem is referenced by:  mptexgf  6601  aciunf1  29693
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