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Theorem uneq1d 3288
Description: Deduction adding union to the right in a class equality. (Contributed by NM, 29-Mar-1998.)
Hypothesis
Ref Expression
uneq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
uneq1d  |-  ( ph  ->  ( A  u.  C
)  =  ( B  u.  C ) )

Proof of Theorem uneq1d
StepHypRef Expression
1 uneq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 uneq1 3282 . 2  |-  ( A  =  B  ->  ( A  u.  C )  =  ( B  u.  C ) )
31, 2syl 14 1  |-  ( ph  ->  ( A  u.  C
)  =  ( B  u.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    u. cun 3127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133
This theorem is referenced by:  ifeq1  3537  preq1  3669  tpeq1  3678  tpeq2  3679  resasplitss  5395  fmptpr  5708  funresdfunsnss  5719  rdgisucinc  6385  oasuc  6464  omsuc  6472  funresdfunsndc  6506  fisseneq  6930  sbthlemi5  6959  exmidfodomrlemim  7199  fzpred  10069  fseq1p1m1  10093  nn0split  10135  nnsplit  10136  fzo0sn0fzo1  10220  fzosplitprm1  10233  fsum1p  11425  fprod1p  11606  zsupcllemstep  11945  setsvala  12492  setsabsd  12500  setscom  12501  prdsex  12717
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