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Theorem tpeq2 4061
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 4052 . . 3  |-  ( A  =  B  ->  { C ,  A }  =  { C ,  B }
)
21uneq1d 3596 . 2  |-  ( A  =  B  ->  ( { C ,  A }  u.  { D } )  =  ( { C ,  B }  u.  { D } ) )
3 df-tp 3977 . 2  |-  { C ,  A ,  D }  =  ( { C ,  A }  u.  { D } )
4 df-tp 3977 . 2  |-  { C ,  B ,  D }  =  ( { C ,  B }  u.  { D } )
52, 3, 43eqtr4g 2468 1  |-  ( A  =  B  ->  { C ,  A ,  D }  =  { C ,  B ,  D } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    u. cun 3412   {csn 3972   {cpr 3974   {ctp 3976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-v 3061  df-un 3419  df-sn 3973  df-pr 3975  df-tp 3977
This theorem is referenced by:  tpeq2d  4064  fztpval  11796  hashtpg  12572  dvh4dimN  34467  lmod1  38604
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