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Theorem iineq2 3903
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 2241 . . . . 5  |-  ( B  =  C  ->  (
y  e.  B  <->  y  e.  C ) )
21ralimi 2540 . . . 4  |-  ( A. x  e.  A  B  =  C  ->  A. x  e.  A  ( y  e.  B  <->  y  e.  C
) )
3 ralbi 2609 . . . 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 2295 . 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 3889 . 2  |-  |^|_ x  e.  A  B  =  { y  |  A. x  e.  A  y  e.  B }
7 df-iin 3889 . 2  |-  |^|_ x  e.  A  C  =  { y  |  A. x  e.  A  y  e.  C }
85, 6, 73eqtr4g 2235 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 105    = wceq 1353    e. wcel 2148   {cab 2163   A.wral 2455   |^|_ciin 3887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-ral 2460  df-iin 3889
This theorem is referenced by:  iineq2i  3905  iineq2d  3906
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