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Theorem ineqan12d 3279
Description: Equality deduction for intersection of two classes. (Contributed by NM, 7-Feb-2007.)
Hypotheses
Ref Expression
ineq1d.1  |-  ( ph  ->  A  =  B )
ineqan12d.2  |-  ( ps 
->  C  =  D
)
Assertion
Ref Expression
ineqan12d  |-  ( (
ph  /\  ps )  ->  ( A  i^i  C
)  =  ( B  i^i  D ) )

Proof of Theorem ineqan12d
StepHypRef Expression
1 ineq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 ineqan12d.2 . 2  |-  ( ps 
->  C  =  D
)
3 ineq12 3272 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  i^i  C
)  =  ( B  i^i  D ) )
41, 2, 3syl2an 287 1  |-  ( (
ph  /\  ps )  ->  ( A  i^i  C
)  =  ( B  i^i  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    i^i cin 3070
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077
This theorem is referenced by:  fvun1  5487  fndmin  5527  offval  5989  ofrfval  5990  offval3  6032  iooinsup  11053
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