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Theorem uniin1 28213
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  |-  U_ x  e.  A  ( x  i^i  B )  =  ( U. A  i^i  B
)
Distinct variable groups:    x, A    x, B

Proof of Theorem uniin1
StepHypRef Expression
1 iunin1 4357 . 2  |-  U_ x  e.  A  ( x  i^i  B )  =  (
U_ x  e.  A  x  i^i  B )
2 uniiun 4345 . . 3  |-  U. A  =  U_ x  e.  A  x
32ineq1i 3642 . 2  |-  ( U. A  i^i  B )  =  ( U_ x  e.  A  x  i^i  B
)
41, 3eqtr4i 2487 1  |-  U_ x  e.  A  ( x  i^i  B )  =  ( U. A  i^i  B
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1455    i^i cin 3415   U.cuni 4212   U_ciun 4292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ral 2754  df-rex 2755  df-v 3059  df-in 3423  df-ss 3430  df-uni 4213  df-iun 4294
This theorem is referenced by: (None)
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