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Theorem ineq2 3177
Description: Equality theorem for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
Assertion
Ref Expression
ineq2  |-  ( A  =  B  ->  ( C  i^i  A )  =  ( C  i^i  B
) )

Proof of Theorem ineq2
StepHypRef Expression
1 ineq1 3176 . 2  |-  ( A  =  B  ->  ( A  i^i  C )  =  ( B  i^i  C
) )
2 incom 3174 . 2  |-  ( C  i^i  A )  =  ( A  i^i  C
)
3 incom 3174 . 2  |-  ( C  i^i  B )  =  ( B  i^i  C
)
41, 2, 33eqtr4g 2140 1  |-  ( A  =  B  ->  ( C  i^i  A )  =  ( C  i^i  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    i^i cin 2981
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-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-in 2988
This theorem is referenced by:  ineq12  3178  ineq2i  3180  ineq2d  3183  uneqin  3231  intprg  3689  uzin2  10092
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