Theorem tpeq123d 4748

Description: Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)

Hypotheses

Ref	Expression
tpeq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)
tpeq123d.2	⊢ (𝜑 → 𝐶 = 𝐷)
tpeq123d.3	⊢ (𝜑 → 𝐸 = 𝐹)

Assertion

Ref	Expression
tpeq123d	⊢ (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐹})

Proof of Theorem tpeq123d

Step	Hyp	Ref	Expression
1		tpeq1d.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2	1	tpeq1d 4745	. 2 ⊢ (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐶, 𝐸})
3		tpeq123d.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
4	3	tpeq2d 4746	. 2 ⊢ (𝜑 → {𝐵, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐸})
5		tpeq123d.3	. . 3 ⊢ (𝜑 → 𝐸 = 𝐹)
6	5	tpeq3d 4747	. 2 ⊢ (𝜑 → {𝐵, 𝐷, 𝐸} = {𝐵, 𝐷, 𝐹})
7	2, 4, 6	3eqtrd 2781	1 ⊢ (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐹})

Syntax hints:   → wi 4   = wceq 1540  {ctp 4630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-sn 4627  df-pr 4629  df-tp 4631

This theorem is referenced by:  fz0tp  13668  fz0to5un2tp  13671  fzo0to3tp  13791  fzo1to4tp  13793  prdsval  17500  imasval  17556  fucval  18006  fucpropd  18025  setcval  18122  catcval  18145  estrcval  18168  xpcval  18222  efmnd  18883  dfcnfldOLD  21380  psrval  21935  om1val  25063  s3rnOLD  32930  rlocval  33263  idlsrgval  33531  ldualset  39126  erngfset  40801  erngfset-rN  40809  dvafset  41006  dvaset  41007  dvhfset  41082  dvhset  41083  hlhilset  41936  rabren3dioph  42826  mendval  43191  oaun3  43395  nnsum4primesodd  47783  nnsum4primesoddALTV  47784  rngcvalALTV  48181  ringcvalALTV  48205  mndtcval  49176