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Theorem tpeq123d 3734
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 3731 . 2  |-  ( ph  ->  { A ,  C ,  E }  =  { B ,  C ,  E } )
3 tpeq123d.2 . . 3  |-  ( ph  ->  C  =  D )
43tpeq2d 3732 . 2  |-  ( ph  ->  { B ,  C ,  E }  =  { B ,  D ,  E } )
5 tpeq123d.3 . . 3  |-  ( ph  ->  E  =  F )
65tpeq3d 3733 . 2  |-  ( ph  ->  { B ,  D ,  E }  =  { B ,  D ,  F } )
72, 4, 63eqtrd 2332 1  |-  ( ph  ->  { A ,  C ,  E }  =  { B ,  D ,  F } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632   {ctp 3655
This theorem is referenced by:  prdsval  13371  imasval  13430  fucval  13848  fucpropd  13867  setcval  13925  catcval  13944  xpcval  13967  symgval  14787  psrval  16126  om1val  18544  rabren3dioph  27001  mendval  27594  fzo0to3tp  28210  usgraexvlem  28261  wlkntrllem3  28347  ldualset  29937  erngfset  31610  erngfset-rN  31618  dvafset  31815  dvaset  31816  dvhfset  31892  dvhset  31893  hlhilset  32749
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-sn 3659  df-pr 3660  df-tp 3661
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