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Theorem ineq1i 3198
Description: Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
Hypothesis
Ref Expression
ineq1i.1  |-  A  =  B
Assertion
Ref Expression
ineq1i  |-  ( A  i^i  C )  =  ( B  i^i  C
)

Proof of Theorem ineq1i
StepHypRef Expression
1 ineq1i.1 . 2  |-  A  =  B
2 ineq1 3195 . 2  |-  ( A  =  B  ->  ( A  i^i  C )  =  ( B  i^i  C
) )
31, 2ax-mp 7 1  |-  ( A  i^i  C )  =  ( B  i^i  C
)
Colors of variables: wff set class
Syntax hints:    = wceq 1290    i^i cin 2999
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-in 3006
This theorem is referenced by:  in12  3212  inindi  3218  dfrab2  3275  dfrab3  3276  disjpr2  3510  resres  4738  imainrect  4889  ssenen  6621  minmax  10722  setsfun  11590  setsfun0  11591
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