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Theorem tpeq123d 3502
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 3499 . 2  |-  ( ph  ->  { A ,  C ,  E }  =  { B ,  C ,  E } )
3 tpeq123d.2 . . 3  |-  ( ph  ->  C  =  D )
43tpeq2d 3500 . 2  |-  ( ph  ->  { B ,  C ,  E }  =  { B ,  D ,  E } )
5 tpeq123d.3 . . 3  |-  ( ph  ->  E  =  F )
65tpeq3d 3501 . 2  |-  ( ph  ->  { B ,  D ,  E }  =  { B ,  D ,  F } )
72, 4, 63eqtrd 2119 1  |-  ( ph  ->  { A ,  C ,  E }  =  { B ,  D ,  F } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285   {ctp 3418
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-sn 3422  df-pr 3423  df-tp 3424
This theorem is referenced by:  fz0tp  9282  fzo0to3tp  9375
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