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Theorem uneq2d 3143
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 3137 . 2 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
31, 2syl 14 1 (𝜑 → (𝐶𝐴) = (𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1287  cun 2986
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-un 2992
This theorem is referenced by:  ifeq2  3383  tpeq3  3515  iununir  3796  unisucg  4217  relcoi1  4930  resasplitss  5155  fvun1  5335  fmptapd  5453  fvunsng  5456  tfr1onlemaccex  6069  tfrcllemaccex  6082  rdgeq1  6092  rdgivallem  6102  rdgisuc1  6105  rdgon  6107  rdg0  6108  oav2  6180  oasuc  6181  omv2  6182  omsuc  6189  unsnfidcex  6584  undifdc  6588  ssfirab  6596  fnfi  6599  sbthlemi5  6617  sbthlemi6  6618  pm54.43  6765  fzsuc  9416  fseq1p1m1  9441  fseq1m1p1  9442  fzosplitsnm1  9551  fzosplitsn  9575  fzosplitprm1  9576  resunimafz0  10136  zfz1isolemsplit  10143  fsumm1  10697
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