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Theorem unundi 3909
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 3891 . . 3 (𝐴𝐴) = 𝐴
21uneq1i 3898 . 2 ((𝐴𝐴) ∪ (𝐵𝐶)) = (𝐴 ∪ (𝐵𝐶))
3 un4 3908 . 2 ((𝐴𝐴) ∪ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
42, 3eqtr3i 2776 1 (𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1624  cun 3705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-v 3334  df-un 3712
This theorem is referenced by:  dfif5  4238
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