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Theorem preq1 3487
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 3427 . . 3 (𝐴 = 𝐵 → {𝐴} = {𝐵})
21uneq1d 3135 . 2 (𝐴 = 𝐵 → ({𝐴} ∪ {𝐶}) = ({𝐵} ∪ {𝐶}))
3 df-pr 3423 . 2 {𝐴, 𝐶} = ({𝐴} ∪ {𝐶})
4 df-pr 3423 . 2 {𝐵, 𝐶} = ({𝐵} ∪ {𝐶})
52, 3, 43eqtr4g 2140 1 (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  cun 2980  {csn 3416  {cpr 3417
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 2612  df-un 2986  df-sn 3422  df-pr 3423
This theorem is referenced by:  preq2  3488  preq12  3489  preq1i  3490  preq1d  3493  tpeq1  3496  prnzg  3532  preq12b  3582  preq12bg  3585  opeq1  3590  uniprg  3636  intprg  3689  prexg  3994  opthreg  4327  bj-prexg  10987
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