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Theorem uniin2 28243
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
uniin2  |-  U_ x  e.  B  ( A  i^i  x )  =  ( A  i^i  U. B
)
Distinct variable groups:    x, A    x, B

Proof of Theorem uniin2
StepHypRef Expression
1 iunin2 4333 . 2  |-  U_ x  e.  B  ( A  i^i  x )  =  ( A  i^i  U_ x  e.  B  x )
2 uniiun 4322 . . 3  |-  U. B  =  U_ x  e.  B  x
32ineq2i 3622 . 2  |-  ( A  i^i  U. B )  =  ( A  i^i  U_ x  e.  B  x )
41, 3eqtr4i 2496 1  |-  U_ x  e.  B  ( A  i^i  x )  =  ( A  i^i  U. B
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1452    i^i cin 3389   U.cuni 4190   U_ciun 4269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-v 3033  df-in 3397  df-uni 4191  df-iun 4271
This theorem is referenced by:  ldgenpisyslem1  29059
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