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Theorem 3unrab 32531
Description: Union of three restricted class abstractions. (Contributed by Thierry Arnoux, 6-Jul-2025.)
Assertion
Ref Expression
3unrab (({𝑥𝐴𝜑} ∪ {𝑥𝐴𝜓}) ∪ {𝑥𝐴𝜒}) = {𝑥𝐴 ∣ (𝜑𝜓𝜒)}

Proof of Theorem 3unrab
StepHypRef Expression
1 unrab 4321 . 2 ({𝑥𝐴 ∣ (𝜑𝜓)} ∪ {𝑥𝐴𝜒}) = {𝑥𝐴 ∣ ((𝜑𝜓) ∨ 𝜒)}
2 unrab 4321 . . 3 ({𝑥𝐴𝜑} ∪ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑𝜓)}
32uneq1i 4174 . 2 (({𝑥𝐴𝜑} ∪ {𝑥𝐴𝜓}) ∪ {𝑥𝐴𝜒}) = ({𝑥𝐴 ∣ (𝜑𝜓)} ∪ {𝑥𝐴𝜒})
4 df-3or 1087 . . 3 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∨ 𝜒))
54rabbii 3439 . 2 {𝑥𝐴 ∣ (𝜑𝜓𝜒)} = {𝑥𝐴 ∣ ((𝜑𝜓) ∨ 𝜒)}
61, 3, 53eqtr4i 2773 1 (({𝑥𝐴𝜑} ∪ {𝑥𝐴𝜓}) ∪ {𝑥𝐴𝜒}) = {𝑥𝐴 ∣ (𝜑𝜓𝜒)}
Colors of variables: wff setvar class
Syntax hints:  wo 847  w3o 1085   = wceq 1537  {crab 3433  cun 3961
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-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-un 3968
This theorem is referenced by:  constrfin  33751
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