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Theorem uneq2d 3289
Description: Deduction adding union to the left in a class equality. (Contributed by NM, 29-Mar-1998.)
Hypothesis
Ref Expression
uneq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
uneq2d (𝜑 → (𝐶𝐴) = (𝐶𝐵))

Proof of Theorem uneq2d
StepHypRef Expression
1 uneq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 uneq2 3283 . 2 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
31, 2syl 14 1 (𝜑 → (𝐶𝐴) = (𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  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:  ifeq2  3538  tpeq3  3680  iununir  3970  unisucg  4414  relcoi1  5160  resasplitss  5395  fvun1  5582  fmptapd  5707  fvunsng  5710  fnsnsplitss  5715  tfr1onlemaccex  6348  tfrcllemaccex  6361  rdgeq1  6371  rdgivallem  6381  rdgisuc1  6384  rdgon  6386  rdg0  6387  oav2  6463  oasuc  6464  omv2  6465  omsuc  6472  fnsnsplitdc  6505  unsnfidcex  6918  undifdc  6922  fiintim  6927  ssfirab  6932  fnfi  6935  fidcenumlemr  6953  sbthlemi5  6959  sbthlemi6  6960  pm54.43  7188  fzsuc  10068  fseq1p1m1  10093  fseq1m1p1  10094  fzosplitsnm1  10208  fzosplitsn  10232  fzosplitprm1  10233  resunimafz0  10810  zfz1isolemsplit  10817  fsumm1  11423  fprodm1  11605  ennnfonelemp1  12406  ennnfonelemhdmp1  12409  ennnfonelemkh  12412  ennnfonelemhf1o  12413  ennnfonelemnn0  12422  strsetsid  12494  setscom  12501  bj-charfundcALT  14531
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