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Theorem uniin1 30297
 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 4986 . 2 𝑥𝐴 (𝑥𝐵) = ( 𝑥𝐴 𝑥𝐵)
2 uniiun 4974 . . 3 𝐴 = 𝑥𝐴 𝑥
32ineq1i 4184 . 2 ( 𝐴𝐵) = ( 𝑥𝐴 𝑥𝐵)
41, 3eqtr4i 2847 1 𝑥𝐴 (𝑥𝐵) = ( 𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1533   ∩ cin 3934  ∪ cuni 4831  ∪ ciun 4911 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-in 3942  df-ss 3951  df-uni 4832  df-iun 4913 This theorem is referenced by: (None)
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