Theorem tpeq123d 4702

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 4699	. 2 ⊢ (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐶, 𝐸})
3		tpeq123d.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
4	3	tpeq2d 4700	. 2 ⊢ (𝜑 → {𝐵, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐸})
5		tpeq123d.3	. . 3 ⊢ (𝜑 → 𝐸 = 𝐹)
6	5	tpeq3d 4701	. 2 ⊢ (𝜑 → {𝐵, 𝐷, 𝐸} = {𝐵, 𝐷, 𝐹})
7	2, 4, 6	3eqtrd 2768	1 ⊢ (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐹})

Syntax hints:   → wi 4   = wceq 1540  {ctp 4583

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3440  df-un 3910  df-sn 4580  df-pr 4582  df-tp 4584

This theorem is referenced by:  fz0tp  13549  fz0to5un2tp  13552  fzo0to3tp  13673  fzo1to4tp  13675  prdsval  17377  imasval  17433  fucval  17886  fucpropd  17905  setcval  18002  catcval  18025  estrcval  18048  xpcval  18101  efmnd  18762  dfcnfldOLD  21295  psrval  21840  om1val  24946  s3rnOLD  32900  rlocval  33212  idlsrgval  33453  ldualset  39106  erngfset  40781  erngfset-rN  40789  dvafset  40986  dvaset  40987  dvhfset  41062  dvhset  41063  hlhilset  41916  rabren3dioph  42791  mendval  43155  oaun3  43358  nnsum4primesodd  47784  nnsum4primesoddALTV  47785  rngcvalALTV  48253  ringcvalALTV  48277  mndtcval  49568