Theorem tpeq123d 4681

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 4678	. 2 ⊢ (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐶, 𝐸})
3		tpeq123d.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
4	3	tpeq2d 4679	. 2 ⊢ (𝜑 → {𝐵, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐸})
5		tpeq123d.3	. . 3 ⊢ (𝜑 → 𝐸 = 𝐹)
6	5	tpeq3d 4680	. 2 ⊢ (𝜑 → {𝐵, 𝐷, 𝐸} = {𝐵, 𝐷, 𝐹})
7	2, 4, 6	3eqtrd 2782	1 ⊢ (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐹})

Syntax hints:   → wi 4   = wceq 1539  {ctp 4562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-un 3888  df-sn 4559  df-pr 4561  df-tp 4563

This theorem is referenced by:  fz0tp  13286  fz0to4untppr  13288  fzo0to3tp  13401  fzo1to4tp  13403  prdsval  17083  imasval  17139  fucval  17591  fucpropd  17611  setcval  17708  catcval  17731  estrcval  17756  xpcval  17810  efmnd  18424  psrval  21028  om1val  24099  s3rn  31122  idlsrgval  31550  ldualset  37066  erngfset  38740  erngfset-rN  38748  dvafset  38945  dvaset  38946  dvhfset  39021  dvhset  39022  hlhilset  39875  rabren3dioph  40553  mendval  40924  nnsum4primesodd  45136  nnsum4primesoddALTV  45137  rngcvalALTV  45407  ringcvalALTV  45453  mndtcval  46252