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Theorem ineq12d 3205
Description: Equality deduction for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Hypotheses
Ref Expression
ineq1d.1  |-  ( ph  ->  A  =  B )
ineq12d.2  |-  ( ph  ->  C  =  D )
Assertion
Ref Expression
ineq12d  |-  ( ph  ->  ( A  i^i  C
)  =  ( B  i^i  D ) )

Proof of Theorem ineq12d
StepHypRef Expression
1 ineq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 ineq12d.2 . 2  |-  ( ph  ->  C  =  D )
3 ineq12 3199 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  i^i  C
)  =  ( B  i^i  D ) )
41, 2, 3syl2anc 404 1  |-  ( ph  ->  ( A  i^i  C
)  =  ( B  i^i  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1290    i^i cin 3001
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 2624  df-in 3008
This theorem is referenced by:  csbing  3210  funprg  5079  funtpg  5080  offval  5879  ofrfval  5880  undifdc  6690  djudom  6839  ressid2  11616  ressval2  11617
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