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Theorem uniin1 32564
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 5072 . 2 𝑥𝐴 (𝑥𝐵) = ( 𝑥𝐴 𝑥𝐵)
2 uniiun 5058 . . 3 𝐴 = 𝑥𝐴 𝑥
32ineq1i 4216 . 2 ( 𝐴𝐵) = ( 𝑥𝐴 𝑥𝐵)
41, 3eqtr4i 2768 1 𝑥𝐴 (𝑥𝐵) = ( 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  cin 3950   cuni 4907   ciun 4991
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-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-in 3958  df-ss 3968  df-uni 4908  df-iun 4993
This theorem is referenced by: (None)
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