MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tpeq1 Structured version   Visualization version   GIF version

Theorem tpeq1 4247
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 4238 . . 3 (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})
21uneq1d 3744 . 2 (𝐴 = 𝐵 → ({𝐴, 𝐶} ∪ {𝐷}) = ({𝐵, 𝐶} ∪ {𝐷}))
3 df-tp 4153 . 2 {𝐴, 𝐶, 𝐷} = ({𝐴, 𝐶} ∪ {𝐷})
4 df-tp 4153 . 2 {𝐵, 𝐶, 𝐷} = ({𝐵, 𝐶} ∪ {𝐷})
52, 3, 43eqtr4g 2680 1 (𝐴 = 𝐵 → {𝐴, 𝐶, 𝐷} = {𝐵, 𝐶, 𝐷})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  cun 3553  {csn 4148  {cpr 4150  {ctp 4152
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 3188  df-un 3560  df-sn 4149  df-pr 4151  df-tp 4153
This theorem is referenced by:  tpeq1d  4250  hashtpg  13205  erngset  35565  erngset-rN  35573  dvh4dimN  36213  lmod1  41566
  Copyright terms: Public domain W3C validator