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Theorem tpeq3 4034
Description: Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
Assertion
Ref Expression
tpeq3  |-  ( A  =  B  ->  { C ,  D ,  A }  =  { C ,  D ,  B } )

Proof of Theorem tpeq3
StepHypRef Expression
1 sneq 3954 . . 3  |-  ( A  =  B  ->  { A }  =  { B } )
21uneq2d 3572 . 2  |-  ( A  =  B  ->  ( { C ,  D }  u.  { A } )  =  ( { C ,  D }  u.  { B } ) )
3 df-tp 3949 . 2  |-  { C ,  D ,  A }  =  ( { C ,  D }  u.  { A } )
4 df-tp 3949 . 2  |-  { C ,  D ,  B }  =  ( { C ,  D }  u.  { B } )
52, 3, 43eqtr4g 2448 1  |-  ( A  =  B  ->  { C ,  D ,  A }  =  { C ,  D ,  B } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1399    u. cun 3387   {csn 3944   {cpr 3946   {ctp 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-v 3036  df-un 3394  df-sn 3945  df-tp 3949
This theorem is referenced by:  tpeq3d  4037  tppreq3  4049  fztpval  11663  hashtpg  12427  dvh4dimN  37587
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