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Theorem rabtru 3692
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 1541 . . 3
2 nfcv 2903 . . . 4 𝑥𝑦
3 rabtru.1 . . . 4 𝑥𝐴
4 nftru 1801 . . . 4 𝑥
5 biidd 262 . . . 4 (𝑥 = 𝑦 → (⊤ ↔ ⊤))
62, 3, 4, 5elrabf 3691 . . 3 (𝑦 ∈ {𝑥𝐴 ∣ ⊤} ↔ (𝑦𝐴 ∧ ⊤))
71, 6mpbiran2 710 . 2 (𝑦 ∈ {𝑥𝐴 ∣ ⊤} ↔ 𝑦𝐴)
87eqriv 2732 1 {𝑥𝐴 ∣ ⊤} = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wtru 1538  wcel 2106  wnfc 2888  {crab 3433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rab 3434  df-v 3480
This theorem is referenced by:  mptexgf  7242  aciunf1  32680
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