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Theorem coeq12d 4528
Description: Equality deduction for composition of two classes. (Contributed by FL, 7-Jun-2012.)
Hypotheses
Ref Expression
coeq12d.1  |-  ( ph  ->  A  =  B )
coeq12d.2  |-  ( ph  ->  C  =  D )
Assertion
Ref Expression
coeq12d  |-  ( ph  ->  ( A  o.  C
)  =  ( B  o.  D ) )

Proof of Theorem coeq12d
StepHypRef Expression
1 coeq12d.1 . . 3  |-  ( ph  ->  A  =  B )
21coeq1d 4525 . 2  |-  ( ph  ->  ( A  o.  C
)  =  ( B  o.  C ) )
3 coeq12d.2 . . 3  |-  ( ph  ->  C  =  D )
43coeq2d 4526 . 2  |-  ( ph  ->  ( B  o.  C
)  =  ( B  o.  D ) )
52, 4eqtrd 2114 1  |-  ( ph  ->  ( A  o.  C
)  =  ( B  o.  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    o. ccom 4375
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 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-in 2980  df-ss 2987  df-br 3794  df-opab 3848  df-co 4380
This theorem is referenced by: (None)
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