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Theorem 3unrab 32522
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 4315 . 2 ({𝑥𝐴 ∣ (𝜑𝜓)} ∪ {𝑥𝐴𝜒}) = {𝑥𝐴 ∣ ((𝜑𝜓) ∨ 𝜒)}
2 unrab 4315 . . 3 ({𝑥𝐴𝜑} ∪ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑𝜓)}
32uneq1i 4164 . 2 (({𝑥𝐴𝜑} ∪ {𝑥𝐴𝜓}) ∪ {𝑥𝐴𝜒}) = ({𝑥𝐴 ∣ (𝜑𝜓)} ∪ {𝑥𝐴𝜒})
4 df-3or 1088 . . 3 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∨ 𝜒))
54rabbii 3442 . 2 {𝑥𝐴 ∣ (𝜑𝜓𝜒)} = {𝑥𝐴 ∣ ((𝜑𝜓) ∨ 𝜒)}
61, 3, 53eqtr4i 2775 1 (({𝑥𝐴𝜑} ∪ {𝑥𝐴𝜓}) ∪ {𝑥𝐴𝜒}) = {𝑥𝐴 ∣ (𝜑𝜓𝜒)}
Colors of variables: wff setvar class
Syntax hints:  wo 848  w3o 1086   = wceq 1540  {crab 3436  cun 3949
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-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-un 3956
This theorem is referenced by:  constrfin  33787
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