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Theorem tpeq1 3662
Description: Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
Assertion
Ref Expression
tpeq1 (𝐴 = 𝐵 → {𝐴, 𝐶, 𝐷} = {𝐵, 𝐶, 𝐷})

Proof of Theorem tpeq1
StepHypRef Expression
1 preq1 3653 . . 3 (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})
21uneq1d 3275 . 2 (𝐴 = 𝐵 → ({𝐴, 𝐶} ∪ {𝐷}) = ({𝐵, 𝐶} ∪ {𝐷}))
3 df-tp 3584 . 2 {𝐴, 𝐶, 𝐷} = ({𝐴, 𝐶} ∪ {𝐷})
4 df-tp 3584 . 2 {𝐵, 𝐶, 𝐷} = ({𝐵, 𝐶} ∪ {𝐷})
52, 3, 43eqtr4g 2224 1 (𝐴 = 𝐵 → {𝐴, 𝐶, 𝐷} = {𝐵, 𝐶, 𝐷})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1343  cun 3114  {csn 3576  {cpr 3577  {ctp 3578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147
This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-tp 3584
This theorem is referenced by:  tpeq1d  3665
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