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Theorem tpeq2 3610
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 3601 . . 3  |-  ( A  =  B  ->  { C ,  A }  =  { C ,  B }
)
21uneq1d 3229 . 2  |-  ( A  =  B  ->  ( { C ,  A }  u.  { D } )  =  ( { C ,  B }  u.  { D } ) )
3 df-tp 3535 . 2  |-  { C ,  A ,  D }  =  ( { C ,  A }  u.  { D } )
4 df-tp 3535 . 2  |-  { C ,  B ,  D }  =  ( { C ,  B }  u.  { D } )
52, 3, 43eqtr4g 2197 1  |-  ( A  =  B  ->  { C ,  A ,  D }  =  { C ,  B ,  D } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    u. cun 3069   {csn 3527   {cpr 3528   {ctp 3529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-tp 3535
This theorem is referenced by:  tpeq2d  3613  fztpval  9875
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