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Theorem iineq2 4102
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 2496 . . . . 5  |-  ( B  =  C  ->  (
y  e.  B  <->  y  e.  C ) )
21ralimi 2773 . . . 4  |-  ( A. x  e.  A  B  =  C  ->  A. x  e.  A  ( y  e.  B  <->  y  e.  C
) )
3 ralbi 2834 . . . 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 16 . . 3  |-  ( A. x  e.  A  B  =  C  ->  ( A. x  e.  A  y  e.  B  <->  A. x  e.  A  y  e.  C )
)
54abbidv 2549 . 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 4088 . 2  |-  |^|_ x  e.  A  B  =  { y  |  A. x  e.  A  y  e.  B }
7 df-iin 4088 . 2  |-  |^|_ x  e.  A  C  =  { y  |  A. x  e.  A  y  e.  C }
85, 6, 73eqtr4g 2492 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 177    = wceq 1652    e. wcel 1725   {cab 2421   A.wral 2697   |^|_ciin 4086
This theorem is referenced by:  iineq2i  4104  iineq2d  4105  firest  13652  iincld  17095  elrfirn2  26741
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-ral 2702  df-iin 4088
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