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Theorem uniin1 29705
 Description: Union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Thierry Arnoux, 21-Jun-2020.)
Assertion
Ref Expression
uniin1 𝑥𝐴 (𝑥𝐵) = ( 𝐴𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem uniin1
StepHypRef Expression
1 iunin1 4719 . 2 𝑥𝐴 (𝑥𝐵) = ( 𝑥𝐴 𝑥𝐵)
2 uniiun 4707 . . 3 𝐴 = 𝑥𝐴 𝑥
32ineq1i 3961 . 2 ( 𝐴𝐵) = ( 𝑥𝐴 𝑥𝐵)
41, 3eqtr4i 2796 1 𝑥𝐴 (𝑥𝐵) = ( 𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1631   ∩ cin 3722  ∪ cuni 4574  ∪ ciun 4654 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-in 3730  df-ss 3737  df-uni 4575  df-iun 4656 This theorem is referenced by: (None)
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