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Theorem tpeq3 3896
Description: Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
Assertion
Ref Expression
tpeq3  |-  ( A  =  B  ->  { C ,  D ,  A }  =  { C ,  D ,  B } )

Proof of Theorem tpeq3
StepHypRef Expression
1 sneq 3827 . . 3  |-  ( A  =  B  ->  { A }  =  { B } )
21uneq2d 3503 . 2  |-  ( A  =  B  ->  ( { C ,  D }  u.  { A } )  =  ( { C ,  D }  u.  { B } ) )
3 df-tp 3824 . 2  |-  { C ,  D ,  A }  =  ( { C ,  D }  u.  { A } )
4 df-tp 3824 . 2  |-  { C ,  D ,  B }  =  ( { C ,  D }  u.  { B } )
52, 3, 43eqtr4g 2495 1  |-  ( A  =  B  ->  { C ,  D ,  A }  =  { C ,  D ,  B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    u. cun 3320   {csn 3816   {cpr 3817   {ctp 3818
This theorem is referenced by:  tpeq3d  3899  tppreq3  3911  fztpval  11109  hashtpg  11693  dvh4dimN  32307
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-un 3327  df-sn 3822  df-tp 3824
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