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Theorem iinin1 4154
Description: Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4137 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 4153 . 2  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  ( B  i^i  C )  =  ( B  i^i  |^|_ x  e.  A  C ) )
2 incom 3525 . . . 4  |-  ( C  i^i  B )  =  ( B  i^i  C
)
32a1i 11 . . 3  |-  ( x  e.  A  ->  ( C  i^i  B )  =  ( B  i^i  C
) )
43iineq2i 4104 . 2  |-  |^|_ x  e.  A  ( C  i^i  B )  =  |^|_ x  e.  A  ( B  i^i  C )
5 incom 3525 . 2  |-  ( |^|_ x  e.  A  C  i^i  B )  =  ( B  i^i  |^|_ x  e.  A  C )
61, 4, 53eqtr4g 2492 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 1652    e. wcel 1725    =/= wne 2598    i^i cin 3311   (/)c0 3620   |^|_ciin 4086
This theorem is referenced by:  firest  13652
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-v 2950  df-dif 3315  df-in 3319  df-nul 3621  df-iin 4088
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