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Theorem tpeq123d 3721
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
StepHypRef Expression
1 tpeq1d.1 . . 3  |-  ( ph  ->  A  =  B )
21tpeq1d 3718 . 2  |-  ( ph  ->  { A ,  C ,  E }  =  { B ,  C ,  E } )
3 tpeq123d.2 . . 3  |-  ( ph  ->  C  =  D )
43tpeq2d 3719 . 2  |-  ( ph  ->  { B ,  C ,  E }  =  { B ,  D ,  E } )
5 tpeq123d.3 . . 3  |-  ( ph  ->  E  =  F )
65tpeq3d 3720 . 2  |-  ( ph  ->  { B ,  D ,  E }  =  { B ,  D ,  F } )
72, 4, 63eqtrd 2319 1  |-  ( ph  ->  { A ,  C ,  E }  =  { B ,  D ,  F } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   {ctp 3642
This theorem is referenced by:  prdsval  13355  imasval  13414  fucval  13832  fucpropd  13851  setcval  13909  catcval  13928  xpcval  13951  symgval  14771  psrval  16110  om1val  18528  rabren3dioph  26898  mendval  27491  usgraexvlem  28127  ldualset  29315  erngfset  30988  erngfset-rN  30996  dvafset  31193  dvaset  31194  dvhfset  31270  dvhset  31271  hlhilset  32127
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-un 3157  df-sn 3646  df-pr 3647  df-tp 3648
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