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Theorem coeq1d 4765
Description: Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
Hypothesis
Ref Expression
coeq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
coeq1d  |-  ( ph  ->  ( A  o.  C
)  =  ( B  o.  C ) )

Proof of Theorem coeq1d
StepHypRef Expression
1 coeq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 coeq1 4761 . 2  |-  ( A  =  B  ->  ( A  o.  C )  =  ( B  o.  C ) )
31, 2syl 14 1  |-  ( ph  ->  ( A  o.  C
)  =  ( B  o.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1343    o. ccom 4608
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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147
This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-in 3122  df-ss 3129  df-br 3983  df-opab 4044  df-co 4613
This theorem is referenced by:  coeq12d  4768  fcof1o  5757  mapen  6812  hashfacen  10749
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