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Theorem uneq2d 3230
Description: Deduction adding union to the left in a class equality. (Contributed by NM, 29-Mar-1998.)
Hypothesis
Ref Expression
uneq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
uneq2d  |-  ( ph  ->  ( C  u.  A
)  =  ( C  u.  B ) )

Proof of Theorem uneq2d
StepHypRef Expression
1 uneq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 uneq2 3224 . 2  |-  ( A  =  B  ->  ( C  u.  A )  =  ( C  u.  B ) )
31, 2syl 14 1  |-  ( ph  ->  ( C  u.  A
)  =  ( C  u.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    u. cun 3069
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-un 3075
This theorem is referenced by:  ifeq2  3478  tpeq3  3611  iununir  3896  unisucg  4336  relcoi1  5070  resasplitss  5302  fvun1  5487  fmptapd  5611  fvunsng  5614  fnsnsplitss  5619  tfr1onlemaccex  6245  tfrcllemaccex  6258  rdgeq1  6268  rdgivallem  6278  rdgisuc1  6281  rdgon  6283  rdg0  6284  oav2  6359  oasuc  6360  omv2  6361  omsuc  6368  fnsnsplitdc  6401  unsnfidcex  6808  undifdc  6812  fiintim  6817  ssfirab  6822  fnfi  6825  fidcenumlemr  6843  sbthlemi5  6849  sbthlemi6  6850  pm54.43  7046  fzsuc  9856  fseq1p1m1  9881  fseq1m1p1  9882  fzosplitsnm1  9993  fzosplitsn  10017  fzosplitprm1  10018  resunimafz0  10581  zfz1isolemsplit  10588  fsumm1  11192  ennnfonelemp1  11926  ennnfonelemhdmp1  11929  ennnfonelemkh  11932  ennnfonelemhf1o  11933  ennnfonelemnn0  11942  strsetsid  12002  setscom  12009
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