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Theorem tpeq2 3457
 Description: Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
Assertion
Ref Expression
tpeq2 (𝐴 = 𝐵 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷})

Proof of Theorem tpeq2
StepHypRef Expression
1 preq2 3448 . . 3 (𝐴 = 𝐵 → {𝐶, 𝐴} = {𝐶, 𝐵})
21uneq1d 3096 . 2 (𝐴 = 𝐵 → ({𝐶, 𝐴} ∪ {𝐷}) = ({𝐶, 𝐵} ∪ {𝐷}))
3 df-tp 3383 . 2 {𝐶, 𝐴, 𝐷} = ({𝐶, 𝐴} ∪ {𝐷})
4 df-tp 3383 . 2 {𝐶, 𝐵, 𝐷} = ({𝐶, 𝐵} ∪ {𝐷})
52, 3, 43eqtr4g 2097 1 (𝐴 = 𝐵 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷})
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ∪ cun 2915  {csn 3375  {cpr 3376  {ctp 3377 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  df-tp 3383 This theorem is referenced by:  tpeq2d  3460  fztpval  8945
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