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Theorem tpeq3 4068
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 3990 . . 3  |-  ( A  =  B  ->  { A }  =  { B } )
21uneq2d 3613 . 2  |-  ( A  =  B  ->  ( { C ,  D }  u.  { A } )  =  ( { C ,  D }  u.  { B } ) )
3 df-tp 3985 . 2  |-  { C ,  D ,  A }  =  ( { C ,  D }  u.  { A } )
4 df-tp 3985 . 2  |-  { C ,  D ,  B }  =  ( { C ,  D }  u.  { B } )
52, 3, 43eqtr4g 2518 1  |-  ( A  =  B  ->  { C ,  D ,  A }  =  { C ,  D ,  B } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    u. cun 3429   {csn 3980   {cpr 3982   {ctp 3984
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 1954  ax-ext 2431
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 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-v 3074  df-un 3436  df-sn 3981  df-tp 3985
This theorem is referenced by:  tpeq3d  4071  tppreq3  4083  fztpval  11630  hashtpg  12299  dvh4dimN  35411
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