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Theorem ineq12 3178
Description: Equality theorem for intersection of two classes. (Contributed by NM, 8-May-1994.)
Assertion
Ref Expression
ineq12  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  i^i  C
)  =  ( B  i^i  D ) )

Proof of Theorem ineq12
StepHypRef Expression
1 ineq1 3176 . 2  |-  ( A  =  B  ->  ( A  i^i  C )  =  ( B  i^i  C
) )
2 ineq2 3177 . 2  |-  ( C  =  D  ->  ( B  i^i  C )  =  ( B  i^i  D
) )
31, 2sylan9eq 2135 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  i^i  C
)  =  ( B  i^i  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = 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:  ineq12i  3181  ineq12d  3184  ineqan12d  3185  fnun  5056  endisj  6390  pm54.43  6588  bj-inex  10983
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