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Theorem tpeq3 4061
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 3977 . . 3  |-  ( A  =  B  ->  { A }  =  { B } )
21uneq2d 3587 . 2  |-  ( A  =  B  ->  ( { C ,  D }  u.  { A } )  =  ( { C ,  D }  u.  { B } ) )
3 df-tp 3972 . 2  |-  { C ,  D ,  A }  =  ( { C ,  D }  u.  { A } )
4 df-tp 3972 . 2  |-  { C ,  D ,  B }  =  ( { C ,  D }  u.  { B } )
52, 3, 43eqtr4g 2509 1  |-  ( A  =  B  ->  { C ,  D ,  A }  =  { C ,  D ,  B } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1443    u. cun 3401   {csn 3967   {cpr 3969   {ctp 3971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-v 3046  df-un 3408  df-sn 3968  df-tp 3972
This theorem is referenced by:  tpeq3d  4064  tppreq3  4076  fztpval  11854  hashtpg  12638  dvh4dimN  35009
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