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Theorem tpeq1 4074
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 4065 . . 3  |-  ( A  =  B  ->  { A ,  C }  =  { B ,  C }
)
21uneq1d 3620 . 2  |-  ( A  =  B  ->  ( { A ,  C }  u.  { D } )  =  ( { B ,  C }  u.  { D } ) )
3 df-tp 3993 . 2  |-  { A ,  C ,  D }  =  ( { A ,  C }  u.  { D } )
4 df-tp 3993 . 2  |-  { B ,  C ,  D }  =  ( { B ,  C }  u.  { D } )
52, 3, 43eqtr4g 2520 1  |-  ( A  =  B  ->  { A ,  C ,  D }  =  { B ,  C ,  D } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    u. cun 3437   {csn 3988   {cpr 3990   {ctp 3992
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 1955  ax-ext 2432
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 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080  df-un 3444  df-sn 3989  df-pr 3991  df-tp 3993
This theorem is referenced by:  tpeq1d  4077  hashtpg  12308  lmod1  31189  erngset  34807  erngset-rN  34815  dvh4dimN  35455
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