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Theorem preq1 3474
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 3413 . . 3 (𝐴 = 𝐵 → {𝐴} = {𝐵})
21uneq1d 3123 . 2 (𝐴 = 𝐵 → ({𝐴} ∪ {𝐶}) = ({𝐵} ∪ {𝐶}))
3 df-pr 3409 . 2 {𝐴, 𝐶} = ({𝐴} ∪ {𝐶})
4 df-pr 3409 . 2 {𝐵, 𝐶} = ({𝐵} ∪ {𝐶})
52, 3, 43eqtr4g 2113 1 (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  cun 2942  {csn 3402  {cpr 3403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-sn 3408  df-pr 3409
This theorem is referenced by:  preq2  3475  preq12  3476  preq1i  3477  preq1d  3480  tpeq1  3483  prnzg  3519  preq12b  3568  preq12bg  3571  opeq1  3576  uniprg  3622  intprg  3675  prexgOLD  3973  prexg  3974  opthreg  4307  bj-prexg  10390
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