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Theorem uneq1 4090
Description: Equality theorem for the union of two classes. (Contributed by NM, 15-Jul-1993.)
Assertion
Ref Expression
uneq1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Proof of Theorem uneq1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eleq2 2827 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
21orbi1d 914 . . 3 (𝐴 = 𝐵 → ((𝑥𝐴𝑥𝐶) ↔ (𝑥𝐵𝑥𝐶)))
3 elun 4083 . . 3 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
4 elun 4083 . . 3 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
52, 3, 43bitr4g 314 . 2 (𝐴 = 𝐵 → (𝑥 ∈ (𝐴𝐶) ↔ 𝑥 ∈ (𝐵𝐶)))
65eqrdv 2736 1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 844   = wceq 1539  wcel 2106  cun 3885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-un 3892
This theorem is referenced by:  uneq2  4091  uneq12  4092  uneq1i  4093  uneq1d  4096  unineq  4211  prprc1  4701  uniprgOLD  4859  relresfld  6179  unexb  7598  oarec  8393  xpider  8577  ralxpmap  8684  undifixp  8722  findcard2  8947  pwfilem  8960  unxpdom  9030  enp1ilem  9051  findcard2OLD  9056  domunfican  9087  pwfilemOLD  9113  fin1a2lem10  10165  incexclem  15548  lcmfunsnlem  16346  ramub1lem1  16727  ramub1  16729  mreexexlem3d  17355  mreexexlem4d  17356  ipodrsima  18259  mplsubglem  21205  mretopd  22243  iscldtop  22246  nconnsubb  22574  plyval  25354  spanun  29907  difeq  30865  unelldsys  32126  isros  32136  unelros  32139  difelros  32140  rossros  32148  measun  32179  inelcarsg  32278  actfunsnf1o  32584  actfunsnrndisj  32585  mrsubvrs  33484  altopthsn  34263  rankung  34468  poimirlem28  35805  islshp  36993  lshpset2N  37133  paddval  37812  nacsfix  40534  eldioph4b  40633  eldioph4i  40634  diophren  40635  clsk3nimkb  41650  isotone1  41658  fiiuncl  42613  founiiun0  42728  infxrpnf  42986  meadjun  44000  hoidmvle  44138
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