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Theorem tpeq1 3715
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 3706 . . 3  |-  ( A  =  B  ->  { A ,  C }  =  { B ,  C }
)
21uneq1d 3328 . 2  |-  ( A  =  B  ->  ( { A ,  C }  u.  { D } )  =  ( { B ,  C }  u.  { D } ) )
3 df-tp 3648 . 2  |-  { A ,  C ,  D }  =  ( { A ,  C }  u.  { D } )
4 df-tp 3648 . 2  |-  { B ,  C ,  D }  =  ( { B ,  C }  u.  { D } )
52, 3, 43eqtr4g 2340 1  |-  ( A  =  B  ->  { A ,  C ,  D }  =  { B ,  C ,  D } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    u. cun 3150   {csn 3640   {cpr 3641   {ctp 3642
This theorem is referenced by:  tpeq1d  3718  tpssg  24932  isibg2aa  26112  isibg2aalem1  26113  erngset  30989  erngset-rN  30997  dvh4dimN  31637
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-un 3157  df-sn 3646  df-pr 3647  df-tp 3648
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