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Theorem iinin1 3973
Description: Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 3956 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 3972 . 2  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  ( B  i^i  C )  =  ( B  i^i  |^|_ x  e.  A  C ) )
2 incom 3361 . . . 4  |-  ( C  i^i  B )  =  ( B  i^i  C
)
32a1i 10 . . 3  |-  ( x  e.  A  ->  ( C  i^i  B )  =  ( B  i^i  C
) )
43iineq2i 3924 . 2  |-  |^|_ x  e.  A  ( C  i^i  B )  =  |^|_ x  e.  A  ( B  i^i  C )
5 incom 3361 . 2  |-  ( |^|_ x  e.  A  C  i^i  B )  =  ( B  i^i  |^|_ x  e.  A  C )
61, 4, 53eqtr4g 2340 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 4    = wceq 1623    e. wcel 1684    =/= wne 2446    i^i cin 3151   (/)c0 3455   |^|_ciin 3906
This theorem is referenced by:  firest  13337
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-v 2790  df-dif 3155  df-in 3159  df-nul 3456  df-iin 3908
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