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Theorem ineq2 3239
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 3238 . 2  |-  ( A  =  B  ->  ( A  i^i  C )  =  ( B  i^i  C
) )
2 incom 3236 . 2  |-  ( C  i^i  A )  =  ( A  i^i  C
)
3 incom 3236 . 2  |-  ( C  i^i  B )  =  ( B  i^i  C
)
41, 2, 33eqtr4g 2173 1  |-  ( A  =  B  ->  ( C  i^i  A )  =  ( C  i^i  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314    i^i cin 3038
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-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  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-nfc 2245  df-v 2660  df-in 3045
This theorem is referenced by:  ineq12  3240  ineq2i  3242  ineq2d  3245  uneqin  3295  intprg  3772  fiintim  6783  uzin2  10699  inopn  12065  basis1  12109  basis2  12110  baspartn  12112  metreslem  12444  qtopbasss  12585
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