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Theorem tpeq2 4255
 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 4246 . . 3 (𝐴 = 𝐵 → {𝐶, 𝐴} = {𝐶, 𝐵})
21uneq1d 3750 . 2 (𝐴 = 𝐵 → ({𝐶, 𝐴} ∪ {𝐷}) = ({𝐶, 𝐵} ∪ {𝐷}))
3 df-tp 4160 . 2 {𝐶, 𝐴, 𝐷} = ({𝐶, 𝐴} ∪ {𝐷})
4 df-tp 4160 . 2 {𝐶, 𝐵, 𝐷} = ({𝐶, 𝐵} ∪ {𝐷})
52, 3, 43eqtr4g 2680 1 (𝐴 = 𝐵 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1480   ∪ cun 3558  {csn 4155  {cpr 4157  {ctp 4159 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3192  df-un 3565  df-sn 4156  df-pr 4158  df-tp 4160 This theorem is referenced by:  tpeq2d  4258  fntpb  6438  fztpval  12360  hashtpg  13221  dvh4dimN  36255  lmod1  41599
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