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Theorem iineq2 3665
Description: Equality theorem for indexed intersection. (Contributed by NM, 22-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iineq2  C  |^|_  |^|_  C

Proof of Theorem iineq2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eleq2 2098 . . . . 5  C  C
21ralimi 2378 . . . 4  C  C
3 ralbi 2439 . . . 4  C  C
42, 3syl 14 . . 3  C  C
54abbidv 2152 . 2  C  {  |  }  {  |  C }
6 df-iin 3651 . 2  |^|_  {  |  }
7 df-iin 3651 . 2  |^|_  C  {  |  C }
85, 6, 73eqtr4g 2094 1  C  |^|_  |^|_  C
Colors of variables: wff set class
Syntax hints:   wi 4   wb 98   wceq 1242   wcel 1390   {cab 2023  wral 2300   |^|_ciin 3649
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-ral 2305  df-iin 3651
This theorem is referenced by:  iineq2i  3667  iineq2d  3668
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