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Theorem tpeq2 4064
Description: Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
Assertion
Ref Expression
tpeq2  |-  ( A  =  B  ->  { C ,  A ,  D }  =  { C ,  B ,  D } )

Proof of Theorem tpeq2
StepHypRef Expression
1 preq2 4055 . . 3  |-  ( A  =  B  ->  { C ,  A }  =  { C ,  B }
)
21uneq1d 3609 . 2  |-  ( A  =  B  ->  ( { C ,  A }  u.  { D } )  =  ( { C ,  B }  u.  { D } ) )
3 df-tp 3982 . 2  |-  { C ,  A ,  D }  =  ( { C ,  A }  u.  { D } )
4 df-tp 3982 . 2  |-  { C ,  B ,  D }  =  ( { C ,  B }  u.  { D } )
52, 3, 43eqtr4g 2517 1  |-  ( A  =  B  ->  { C ,  A ,  D }  =  { C ,  B ,  D } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    u. cun 3426   {csn 3977   {cpr 3979   {ctp 3981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-v 3072  df-un 3433  df-sn 3978  df-pr 3980  df-tp 3982
This theorem is referenced by:  tpeq2d  4067  fztpval  11621  hashtpg  12290  lmod1  31143  dvh4dimN  35400
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