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Theorem uneq1 3131
Description: Equality theorem for union of two classes. (Contributed by NM, 5-Aug-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 2146 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
21orbi1d 738 . . 3 (𝐴 = 𝐵 → ((𝑥𝐴𝑥𝐶) ↔ (𝑥𝐵𝑥𝐶)))
3 elun 3125 . . 3 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
4 elun 3125 . . 3 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
52, 3, 43bitr4g 221 . 2 (𝐴 = 𝐵 → (𝑥 ∈ (𝐴𝐶) ↔ 𝑥 ∈ (𝐵𝐶)))
65eqrdv 2081 1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 662   = wceq 1285  wcel 1434  cun 2982
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-un 2988
This theorem is referenced by:  uneq2  3132  uneq12  3133  uneq1i  3134  uneq1d  3137  prprc1  3524  uniprg  3642  unexb  4231  relresfld  4914  relcoi1  4916  rdgeq2  6069  xpiderm  6293  findcard2  6535  findcard2s  6536  unfiexmid  6555  bdunexb  11154  bj-unexg  11155
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