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Theorem tpeq123d 4683
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 4680 . 2 (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐶, 𝐸})
3 tpeq123d.2 . . 3 (𝜑𝐶 = 𝐷)
43tpeq2d 4681 . 2 (𝜑 → {𝐵, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐸})
5 tpeq123d.3 . . 3 (𝜑𝐸 = 𝐹)
65tpeq3d 4682 . 2 (𝜑 → {𝐵, 𝐷, 𝐸} = {𝐵, 𝐷, 𝐹})
72, 4, 63eqtrd 2865 1 (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐹})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  {ctp 4568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-v 3502  df-un 3945  df-sn 4565  df-pr 4567  df-tp 4569
This theorem is referenced by:  fz0tp  12998  fz0to4untppr  13000  fzo0to3tp  13113  fzo1to4tp  13115  prdsval  16718  imasval  16774  fucval  17218  fucpropd  17237  setcval  17327  catcval  17346  estrcval  17364  xpcval  17417  symgval  18427  psrval  20061  om1val  23549  s3rn  30536  ldualset  36128  erngfset  37802  erngfset-rN  37810  dvafset  38007  dvaset  38008  dvhfset  38083  dvhset  38084  hlhilset  38937  rabren3dioph  39277  mendval  39648  nnsum4primesodd  43793  nnsum4primesoddALTV  43794  rngcvalALTV  44064  ringcvalALTV  44110
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