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Theorem unundi 4131
Description: Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
unundi (𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))

Proof of Theorem unundi
StepHypRef Expression
1 unidm 4113 . . 3 (𝐴𝐴) = 𝐴
21uneq1i 4120 . 2 ((𝐴𝐴) ∪ (𝐵𝐶)) = (𝐴 ∪ (𝐵𝐶))
3 un4 4130 . 2 ((𝐴𝐴) ∪ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
42, 3eqtr3i 2849 1 (𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  cun 3917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-un 3924
This theorem is referenced by:  dfif5  4465
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