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Theorem preq1 3687
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 3621 . . 3 (𝐴 = 𝐵 → {𝐴} = {𝐵})
21uneq1d 3303 . 2 (𝐴 = 𝐵 → ({𝐴} ∪ {𝐶}) = ({𝐵} ∪ {𝐶}))
3 df-pr 3617 . 2 {𝐴, 𝐶} = ({𝐴} ∪ {𝐶})
4 df-pr 3617 . 2 {𝐵, 𝐶} = ({𝐵} ∪ {𝐶})
52, 3, 43eqtr4g 2247 1 (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  cun 3142  {csn 3610  {cpr 3611
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-sn 3616  df-pr 3617
This theorem is referenced by:  preq2  3688  preq12  3689  preq1i  3690  preq1d  3693  tpeq1  3696  prnzg  3734  preq12b  3788  preq12bg  3791  opeq1  3796  uniprg  3842  intprg  3895  prexg  4232  opthreg  4576  bdxmet  14486  bj-prexg  15149
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