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Theorem tpeq1 4121
Description: Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
Assertion
Ref Expression
tpeq1  |-  ( A  =  B  ->  { A ,  C ,  D }  =  { B ,  C ,  D } )

Proof of Theorem tpeq1
StepHypRef Expression
1 preq1 4112 . . 3  |-  ( A  =  B  ->  { A ,  C }  =  { B ,  C }
)
21uneq1d 3662 . 2  |-  ( A  =  B  ->  ( { A ,  C }  u.  { D } )  =  ( { B ,  C }  u.  { D } ) )
3 df-tp 4038 . 2  |-  { A ,  C ,  D }  =  ( { A ,  C }  u.  { D } )
4 df-tp 4038 . 2  |-  { B ,  C ,  D }  =  ( { B ,  C }  u.  { D } )
52, 3, 43eqtr4g 2533 1  |-  ( A  =  B  ->  { A ,  C ,  D }  =  { B ,  C ,  D } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    u. cun 3479   {csn 4033   {cpr 4035   {ctp 4037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3120  df-un 3486  df-sn 4034  df-pr 4036  df-tp 4038
This theorem is referenced by:  tpeq1d  4124  hashtpg  12504  lmod1  32575  erngset  35997  erngset-rN  36005  dvh4dimN  36645
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