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Theorem uneq1d 3289
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 3283 . 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 3128
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 2740  df-un 3134
This theorem is referenced by:  ifeq1  3538  preq1  3670  tpeq1  3679  tpeq2  3680  resasplitss  5396  fmptpr  5709  funresdfunsnss  5720  rdgisucinc  6386  oasuc  6465  omsuc  6473  funresdfunsndc  6507  fisseneq  6931  sbthlemi5  6960  exmidfodomrlemim  7200  fzpred  10070  fseq1p1m1  10094  nn0split  10136  nnsplit  10137  fzo0sn0fzo1  10221  fzosplitprm1  10234  fsum1p  11426  fprod1p  11607  zsupcllemstep  11946  setsvala  12493  setsabsd  12501  setscom  12502  prdsex  12718
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