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Theorem uniin1 29231
 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 4556 . 2 𝑥𝐴 (𝑥𝐵) = ( 𝑥𝐴 𝑥𝐵)
2 uniiun 4544 . . 3 𝐴 = 𝑥𝐴 𝑥
32ineq1i 3793 . 2 ( 𝐴𝐵) = ( 𝑥𝐴 𝑥𝐵)
41, 3eqtr4i 2646 1 𝑥𝐴 (𝑥𝐵) = ( 𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1480   ∩ cin 3558  ∪ cuni 4407  ∪ ciun 4490 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3191  df-in 3566  df-ss 3573  df-uni 4408  df-iun 4492 This theorem is referenced by: (None)
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