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Theorem iinin2 3988
Description: Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 3972 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
Allowed substitution hint:    C( x)

Proof of Theorem iinin2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 r19.28zv 3562 . . . 4  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  (
y  e.  B  /\  y  e.  C )  <->  ( y  e.  B  /\  A. x  e.  A  y  e.  C ) ) )
2 elin 3371 . . . . 5  |-  ( y  e.  ( B  i^i  C )  <->  ( y  e.  B  /\  y  e.  C ) )
32ralbii 2580 . . . 4  |-  ( A. x  e.  A  y  e.  ( B  i^i  C
)  <->  A. x  e.  A  ( y  e.  B  /\  y  e.  C
) )
4 vex 2804 . . . . . 6  |-  y  e. 
_V
5 eliin 3926 . . . . . 6  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  C  <->  A. x  e.  A  y  e.  C ) )
64, 5ax-mp 8 . . . . 5  |-  ( y  e.  |^|_ x  e.  A  C 
<-> 
A. x  e.  A  y  e.  C )
76anbi2i 675 . . . 4  |-  ( ( y  e.  B  /\  y  e.  |^|_ x  e.  A  C )  <->  ( y  e.  B  /\  A. x  e.  A  y  e.  C ) )
81, 3, 73bitr4g 279 . . 3  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  y  e.  ( B  i^i  C
)  <->  ( y  e.  B  /\  y  e. 
|^|_ x  e.  A  C ) ) )
9 eliin 3926 . . . 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 8 . . 3  |-  ( y  e.  |^|_ x  e.  A  ( B  i^i  C )  <->  A. x  e.  A  y  e.  ( B  i^i  C ) )
11 elin 3371 . . 3  |-  ( y  e.  ( B  i^i  |^|_
x  e.  A  C
)  <->  ( y  e.  B  /\  y  e. 
|^|_ x  e.  A  C ) )
128, 10, 113bitr4g 279 . 2  |-  ( A  =/=  (/)  ->  ( y  e.  |^|_ x  e.  A  ( B  i^i  C )  <-> 
y  e.  ( B  i^i  |^|_ x  e.  A  C ) ) )
1312eqrdv 2294 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 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   _Vcvv 2801    i^i cin 3164   (/)c0 3468   |^|_ciin 3922
This theorem is referenced by:  iinin1  3989
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-v 2803  df-dif 3168  df-in 3172  df-nul 3469  df-iin 3924
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