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Theorem tpeq1 3885
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 3876 . . 3  |-  ( A  =  B  ->  { A ,  C }  =  { B ,  C }
)
21uneq1d 3493 . 2  |-  ( A  =  B  ->  ( { A ,  C }  u.  { D } )  =  ( { B ,  C }  u.  { D } ) )
3 df-tp 3815 . 2  |-  { A ,  C ,  D }  =  ( { A ,  C }  u.  { D } )
4 df-tp 3815 . 2  |-  { B ,  C ,  D }  =  ( { B ,  C }  u.  { D } )
52, 3, 43eqtr4g 2493 1  |-  ( A  =  B  ->  { A ,  C ,  D }  =  { B ,  C ,  D } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    u. cun 3311   {csn 3807   {cpr 3808   {ctp 3809
This theorem is referenced by:  tpeq1d  3888  hashtpg  11684  erngset  31535  erngset-rN  31543  dvh4dimN  32183
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2951  df-un 3318  df-sn 3813  df-pr 3814  df-tp 3815
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