ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iinin2m Unicode version

Theorem iinin2m 3780
Description: Indexed intersection of intersection. Compare to Theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Jim Kingdon, 17-Aug-2018.)
Assertion
Ref Expression
iinin2m  |-  ( E. x  x  e.  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 iinin2m
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 r19.28mv 3361 . . . 4  |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  ( y  e.  B  /\  y  e.  C )  <->  ( y  e.  B  /\  A. x  e.  A  y  e.  C ) ) )
2 elin 3172 . . . . 5  |-  ( y  e.  ( B  i^i  C )  <->  ( y  e.  B  /\  y  e.  C ) )
32ralbii 2380 . . . 4  |-  ( A. x  e.  A  y  e.  ( B  i^i  C
)  <->  A. x  e.  A  ( y  e.  B  /\  y  e.  C
) )
4 vex 2618 . . . . . 6  |-  y  e. 
_V
5 eliin 3717 . . . . . 6  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  C  <->  A. x  e.  A  y  e.  C ) )
64, 5ax-mp 7 . . . . 5  |-  ( y  e.  |^|_ x  e.  A  C 
<-> 
A. x  e.  A  y  e.  C )
76anbi2i 445 . . . 4  |-  ( ( y  e.  B  /\  y  e.  |^|_ x  e.  A  C )  <->  ( y  e.  B  /\  A. x  e.  A  y  e.  C ) )
81, 3, 73bitr4g 221 . . 3  |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  y  e.  ( B  i^i  C )  <-> 
( y  e.  B  /\  y  e.  |^|_ x  e.  A  C )
) )
9 eliin 3717 . . . 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 7 . . 3  |-  ( y  e.  |^|_ x  e.  A  ( B  i^i  C )  <->  A. x  e.  A  y  e.  ( B  i^i  C ) )
11 elin 3172 . . 3  |-  ( y  e.  ( B  i^i  |^|_
x  e.  A  C
)  <->  ( y  e.  B  /\  y  e. 
|^|_ x  e.  A  C ) )
128, 10, 113bitr4g 221 . 2  |-  ( E. x  x  e.  A  ->  ( y  e.  |^|_ x  e.  A  ( B  i^i  C )  <->  y  e.  ( B  i^i  |^|_ x  e.  A  C )
) )
1312eqrdv 2083 1  |-  ( E. x  x  e.  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    /\ wa 102    <-> wb 103    = wceq 1287   E.wex 1424    e. wcel 1436   A.wral 2355   _Vcvv 2615    i^i cin 2987   |^|_ciin 3713
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-v 2617  df-in 2994  df-iin 3715
This theorem is referenced by:  iinin1m  3781
  Copyright terms: Public domain W3C validator