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Theorem tpeq2 4109
Description: Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
Assertion
Ref Expression
tpeq2  |-  ( A  =  B  ->  { C ,  A ,  D }  =  { C ,  B ,  D } )

Proof of Theorem tpeq2
StepHypRef Expression
1 preq2 4100 . . 3  |-  ( A  =  B  ->  { C ,  A }  =  { C ,  B }
)
21uneq1d 3650 . 2  |-  ( A  =  B  ->  ( { C ,  A }  u.  { D } )  =  ( { C ,  B }  u.  { D } ) )
3 df-tp 4025 . 2  |-  { C ,  A ,  D }  =  ( { C ,  A }  u.  { D } )
4 df-tp 4025 . 2  |-  { C ,  B ,  D }  =  ( { C ,  B }  u.  { D } )
52, 3, 43eqtr4g 2526 1  |-  ( A  =  B  ->  { C ,  A ,  D }  =  { C ,  B ,  D } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    u. cun 3467   {csn 4020   {cpr 4022   {ctp 4024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-v 3108  df-un 3474  df-sn 4021  df-pr 4023  df-tp 4025
This theorem is referenced by:  tpeq2d  4112  fztpval  11730  hashtpg  12476  lmod1  32049  dvh4dimN  36119
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