Theorem tpeq123d 4114

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 4111	. 2 |- ( ph -> { A , C , E } = { B , C , E } )
3		tpeq123d.2	. . 3 |- ( ph -> C = D )
4	3	tpeq2d 4112	. 2 |- ( ph -> { B , C , E } = { B , D , E } )
5		tpeq123d.3	. . 3 |- ( ph -> E = F )
6	5	tpeq3d 4113	. 2 |- ( ph -> { B , D , E } = { B , D , F } )
7	2, 4, 6	3eqtrd 2505	1 |- ( ph -> { A , C , E } = { B , D , F } )

Syntax hints:   -> wi 4   = wceq 1374   {ctp 4024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-v 3108  df-un 3474  df-sn 4021  df-pr 4023  df-tp 4025

This theorem is referenced by:  fz0tp  11764  fzo0to3tp  11857  prdsval  14699  imasval  14755  fucval  15174  fucpropd  15193  setcval  15251  catcval  15270  xpcval  15293  symgval  16192  psrval  17775  om1val  21258  usgraexvlem  24057  rabren3dioph  30340  mendval  30726  ldualset  33797  erngfset  35470  erngfset-rN  35478  dvafset  35675  dvaset  35676  dvhfset  35752  dvhset  35753  hlhilset  36609