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Theorem preq1 3514
Description: Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.)
Assertion
Ref Expression
preq1 (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})

Proof of Theorem preq1
StepHypRef Expression
1 sneq 3452 . . 3 (𝐴 = 𝐵 → {𝐴} = {𝐵})
21uneq1d 3151 . 2 (𝐴 = 𝐵 → ({𝐴} ∪ {𝐶}) = ({𝐵} ∪ {𝐶}))
3 df-pr 3448 . 2 {𝐴, 𝐶} = ({𝐴} ∪ {𝐶})
4 df-pr 3448 . 2 {𝐵, 𝐶} = ({𝐵} ∪ {𝐶})
52, 3, 43eqtr4g 2145 1 (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  cun 2995  {csn 3441  {cpr 3442
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448
This theorem is referenced by:  preq2  3515  preq12  3516  preq1i  3517  preq1d  3520  tpeq1  3523  prnzg  3559  preq12b  3609  preq12bg  3612  opeq1  3617  uniprg  3663  intprg  3716  prexg  4029  opthreg  4362  bj-prexg  11459
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