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Theorem tpeq2 3729
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 3720 . . 3  |-  ( A  =  B  ->  { C ,  A }  =  { C ,  B }
)
21uneq1d 3341 . 2  |-  ( A  =  B  ->  ( { C ,  A }  u.  { D } )  =  ( { C ,  B }  u.  { D } ) )
3 df-tp 3661 . 2  |-  { C ,  A ,  D }  =  ( { C ,  A }  u.  { D } )
4 df-tp 3661 . 2  |-  { C ,  B ,  D }  =  ( { C ,  B }  u.  { D } )
52, 3, 43eqtr4g 2353 1  |-  ( A  =  B  ->  { C ,  A ,  D }  =  { C ,  B ,  D } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    u. cun 3163   {csn 3653   {cpr 3654   {ctp 3655
This theorem is referenced by:  tpeq2d  3732  fztpval  10861  tpssg  25035  isibg2aa  26215  isibg2aalem1  26216  hashtpg  28217  dvh4dimN  32259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-sn 3659  df-pr 3660  df-tp 3661
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