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Theorem tpeq2 4222
 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 4213 . . 3 (𝐴 = 𝐵 → {𝐶, 𝐴} = {𝐶, 𝐵})
21uneq1d 3728 . 2 (𝐴 = 𝐵 → ({𝐶, 𝐴} ∪ {𝐷}) = ({𝐶, 𝐵} ∪ {𝐷}))
3 df-tp 4130 . 2 {𝐶, 𝐴, 𝐷} = ({𝐶, 𝐴} ∪ {𝐷})
4 df-tp 4130 . 2 {𝐶, 𝐵, 𝐷} = ({𝐶, 𝐵} ∪ {𝐷})
52, 3, 43eqtr4g 2669 1 (𝐴 = 𝐵 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∪ cun 3538  {csn 4125  {cpr 4127  {ctp 4129 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-sn 4126  df-pr 4128  df-tp 4130 This theorem is referenced by:  tpeq2d  4225  fntpb  6378  fztpval  12272  hashtpg  13121  dvh4dimN  35754  lmod1  42075
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