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Theorem preq1 3447
 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 3386 . . 3 (𝐴 = 𝐵 → {𝐴} = {𝐵})
21uneq1d 3096 . 2 (𝐴 = 𝐵 → ({𝐴} ∪ {𝐶}) = ({𝐵} ∪ {𝐶}))
3 df-pr 3382 . 2 {𝐴, 𝐶} = ({𝐴} ∪ {𝐶})
4 df-pr 3382 . 2 {𝐵, 𝐶} = ({𝐵} ∪ {𝐶})
52, 3, 43eqtr4g 2097 1 (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ∪ cun 2915  {csn 3375  {cpr 3376 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382 This theorem is referenced by:  preq2  3448  preq12  3449  preq1i  3450  preq1d  3453  tpeq1  3456  prnzg  3492  preq12b  3541  preq12bg  3544  opeq1  3549  uniprg  3595  intprg  3648  prexgOLD  3946  prexg  3947  opthreg  4280  bj-prexg  10031
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