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Theorem tpeq1 3835
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 3826 . . 3  |-  ( A  =  B  ->  { A ,  C }  =  { B ,  C }
)
21uneq1d 3443 . 2  |-  ( A  =  B  ->  ( { A ,  C }  u.  { D } )  =  ( { B ,  C }  u.  { D } ) )
3 df-tp 3765 . 2  |-  { A ,  C ,  D }  =  ( { A ,  C }  u.  { D } )
4 df-tp 3765 . 2  |-  { B ,  C ,  D }  =  ( { B ,  C }  u.  { D } )
52, 3, 43eqtr4g 2444 1  |-  ( A  =  B  ->  { A ,  C ,  D }  =  { B ,  C ,  D } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    u. cun 3261   {csn 3757   {cpr 3758   {ctp 3759
This theorem is referenced by:  tpeq1d  3838  hashtpg  11618  erngset  30914  erngset-rN  30922  dvh4dimN  31562
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-v 2901  df-un 3268  df-sn 3763  df-pr 3764  df-tp 3765
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