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Theorem iinin2 3973
Description: Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 3957 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)
Assertion
Ref Expression
iinin2  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  ( B  i^i  C )  =  ( B  i^i  |^|_ x  e.  A  C ) )
Distinct variable groups:    x, A    x, B
Dummy variable  y is distinct from all other variables.
Allowed substitution hint:    C( x)

Proof of Theorem iinin2
StepHypRef Expression
1 r19.28zv 3550 . . . 4  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  (
y  e.  B  /\  y  e.  C )  <->  ( y  e.  B  /\  A. x  e.  A  y  e.  C ) ) )
2 elin 3359 . . . . 5  |-  ( y  e.  ( B  i^i  C )  <->  ( y  e.  B  /\  y  e.  C ) )
32ralbii 2568 . . . 4  |-  ( A. x  e.  A  y  e.  ( B  i^i  C
)  <->  A. x  e.  A  ( y  e.  B  /\  y  e.  C
) )
4 vex 2792 . . . . . 6  |-  y  e. 
_V
5 eliin 3911 . . . . . 6  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  C  <->  A. x  e.  A  y  e.  C ) )
64, 5ax-mp 10 . . . . 5  |-  ( y  e.  |^|_ x  e.  A  C 
<-> 
A. x  e.  A  y  e.  C )
76anbi2i 677 . . . 4  |-  ( ( y  e.  B  /\  y  e.  |^|_ x  e.  A  C )  <->  ( y  e.  B  /\  A. x  e.  A  y  e.  C ) )
81, 3, 73bitr4g 281 . . 3  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  y  e.  ( B  i^i  C
)  <->  ( y  e.  B  /\  y  e. 
|^|_ x  e.  A  C ) ) )
9 eliin 3911 . . . 4  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  ( B  i^i  C )  <->  A. x  e.  A  y  e.  ( B  i^i  C ) ) )
104, 9ax-mp 10 . . 3  |-  ( y  e.  |^|_ x  e.  A  ( B  i^i  C )  <->  A. x  e.  A  y  e.  ( B  i^i  C ) )
11 elin 3359 . . 3  |-  ( y  e.  ( B  i^i  |^|_
x  e.  A  C
)  <->  ( y  e.  B  /\  y  e. 
|^|_ x  e.  A  C ) )
128, 10, 113bitr4g 281 . 2  |-  ( A  =/=  (/)  ->  ( y  e.  |^|_ x  e.  A  ( B  i^i  C )  <-> 
y  e.  ( B  i^i  |^|_ x  e.  A  C ) ) )
1312eqrdv 2282 1  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  ( B  i^i  C )  =  ( B  i^i  |^|_ x  e.  A  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1624    e. wcel 1685    =/= wne 2447   A.wral 2544   _Vcvv 2789    i^i cin 3152   (/)c0 3456   |^|_ciin 3907
This theorem is referenced by:  iinin1  3974
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-v 2791  df-dif 3156  df-in 3160  df-nul 3457  df-iin 3909
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