MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tpeq123d Structured version   Visualization version   GIF version

Theorem tpeq123d 4227
Description: Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
Hypotheses
Ref Expression
tpeq1d.1 (𝜑𝐴 = 𝐵)
tpeq123d.2 (𝜑𝐶 = 𝐷)
tpeq123d.3 (𝜑𝐸 = 𝐹)
Assertion
Ref Expression
tpeq123d (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐹})

Proof of Theorem tpeq123d
StepHypRef Expression
1 tpeq1d.1 . . 3 (𝜑𝐴 = 𝐵)
21tpeq1d 4224 . 2 (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐶, 𝐸})
3 tpeq123d.2 . . 3 (𝜑𝐶 = 𝐷)
43tpeq2d 4225 . 2 (𝜑 → {𝐵, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐸})
5 tpeq123d.3 . . 3 (𝜑𝐸 = 𝐹)
65tpeq3d 4226 . 2 (𝜑 → {𝐵, 𝐷, 𝐸} = {𝐵, 𝐷, 𝐹})
72, 4, 63eqtrd 2648 1 (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐹})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  {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:  fz0tp  12309  fz0to4untppr  12311  fzo0to3tp  12421  fzo1to4tp  12423  prdsval  15938  imasval  15994  fucval  16441  fucpropd  16460  setcval  16550  catcval  16569  estrcval  16587  xpcval  16640  symgval  17622  psrval  19183  om1val  22638  ldualset  33430  erngfset  35105  erngfset-rN  35113  dvafset  35310  dvaset  35311  dvhfset  35387  dvhset  35388  hlhilset  36244  rabren3dioph  36397  mendval  36772  nnsum4primesodd  40212  nnsum4primesoddALTV  40213  rngcvalALTV  41753  ringcvalALTV  41799
  Copyright terms: Public domain W3C validator