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Theorem iineq2 3703
Description: Equality theorem for indexed intersection. (Contributed by NM, 22-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iineq2  |-  ( A. x  e.  A  B  =  C  ->  |^|_ x  e.  A  B  =  |^|_
x  e.  A  C
)

Proof of Theorem iineq2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eleq2 2143 . . . . 5  |-  ( B  =  C  ->  (
y  e.  B  <->  y  e.  C ) )
21ralimi 2427 . . . 4  |-  ( A. x  e.  A  B  =  C  ->  A. x  e.  A  ( y  e.  B  <->  y  e.  C
) )
3 ralbi 2490 . . . 4  |-  ( A. x  e.  A  (
y  e.  B  <->  y  e.  C )  ->  ( A. x  e.  A  y  e.  B  <->  A. x  e.  A  y  e.  C ) )
42, 3syl 14 . . 3  |-  ( A. x  e.  A  B  =  C  ->  ( A. x  e.  A  y  e.  B  <->  A. x  e.  A  y  e.  C )
)
54abbidv 2197 . 2  |-  ( A. x  e.  A  B  =  C  ->  { y  |  A. x  e.  A  y  e.  B }  =  { y  |  A. x  e.  A  y  e.  C }
)
6 df-iin 3689 . 2  |-  |^|_ x  e.  A  B  =  { y  |  A. x  e.  A  y  e.  B }
7 df-iin 3689 . 2  |-  |^|_ x  e.  A  C  =  { y  |  A. x  e.  A  y  e.  C }
85, 6, 73eqtr4g 2139 1  |-  ( A. x  e.  A  B  =  C  ->  |^|_ x  e.  A  B  =  |^|_
x  e.  A  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1285    e. wcel 1434   {cab 2068   A.wral 2349   |^|_ciin 3687
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-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-ral 2354  df-iin 3689
This theorem is referenced by:  iineq2i  3705  iineq2d  3706
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