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Theorem iineq2 3798
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 2179 . . . . 5  |-  ( B  =  C  ->  (
y  e.  B  <->  y  e.  C ) )
21ralimi 2470 . . . 4  |-  ( A. x  e.  A  B  =  C  ->  A. x  e.  A  ( y  e.  B  <->  y  e.  C
) )
3 ralbi 2539 . . . 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 2233 . 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 3784 . 2  |-  |^|_ x  e.  A  B  =  { y  |  A. x  e.  A  y  e.  B }
7 df-iin 3784 . 2  |-  |^|_ x  e.  A  C  =  { y  |  A. x  e.  A  y  e.  C }
85, 6, 73eqtr4g 2173 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 104    = wceq 1314    e. wcel 1463   {cab 2101   A.wral 2391   |^|_ciin 3782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-ral 2396  df-iin 3784
This theorem is referenced by:  iineq2i  3800  iineq2d  3801
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