Theorem tpeq123d 4283

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

Hypotheses

Ref	Expression
tpeq1d.1	|- ( ph -> A = B )
tpeq123d.2	|- ( ph -> C = D )
tpeq123d.3	|- ( ph -> E = F )

Assertion

Ref	Expression
tpeq123d	|- ( ph -> { A , C , E } = { B , D , F } )

Proof of Theorem tpeq123d

Step	Hyp	Ref	Expression
1		tpeq1d.1	. . 3 |- ( ph -> A = B )
2	1	tpeq1d 4280	. 2 |- ( ph -> { A , C , E } = { B , C , E } )
3		tpeq123d.2	. . 3 |- ( ph -> C = D )
4	3	tpeq2d 4281	. 2 |- ( ph -> { B , C , E } = { B , D , E } )
5		tpeq123d.3	. . 3 |- ( ph -> E = F )
6	5	tpeq3d 4282	. 2 |- ( ph -> { B , D , E } = { B , D , F } )
7	2, 4, 6	3eqtrd 2660	1 |- ( ph -> { A , C , E } = { B , D , F } )

Syntax hints:   -> wi 4   = wceq 1483   {ctp 4181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-sn 4178  df-pr 4180  df-tp 4182

This theorem is referenced by:  fz0tp  12440  fz0to4untppr  12442  fzo0to3tp  12554  fzo1to4tp  12556  prdsval  16115  imasval  16171  fucval  16618  fucpropd  16637  setcval  16727  catcval  16746  estrcval  16764  xpcval  16817  symgval  17799  psrval  19362  om1val  22830  ldualset  34412  erngfset  36087  erngfset-rN  36095  dvafset  36292  dvaset  36293  dvhfset  36369  dvhset  36370  hlhilset  37226  rabren3dioph  37379  mendval  37753  nnsum4primesodd  41684  nnsum4primesoddALTV  41685  rngcvalALTV  41961  ringcvalALTV  42007