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Theorem iinin1 3914
Description: Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 3897 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)
Assertion
Ref Expression
iinin1  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  ( C  i^i  B )  =  ( |^|_ x  e.  A  C  i^i  B ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    C( x)

Proof of Theorem iinin1
StepHypRef Expression
1 iinin2 3913 . 2  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  ( B  i^i  C )  =  ( B  i^i  |^|_ x  e.  A  C ) )
2 incom 3303 . . . 4  |-  ( C  i^i  B )  =  ( B  i^i  C
)
32a1i 12 . . 3  |-  ( x  e.  A  ->  ( C  i^i  B )  =  ( B  i^i  C
) )
43iineq2i 3865 . 2  |-  |^|_ x  e.  A  ( C  i^i  B )  =  |^|_ x  e.  A  ( B  i^i  C )
5 incom 3303 . 2  |-  ( |^|_ x  e.  A  C  i^i  B )  =  ( B  i^i  |^|_ x  e.  A  C )
61, 4, 53eqtr4g 2313 1  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  ( C  i^i  B )  =  ( |^|_ x  e.  A  C  i^i  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1619    e. wcel 1621    =/= wne 2419    i^i cin 3093   (/)c0 3397   |^|_ciin 3847
This theorem is referenced by:  firest  13264
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-v 2742  df-dif 3097  df-in 3101  df-nul 3398  df-iin 3849
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