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Theorem tppreq3 4121
Description: An unordered triple is an unordered pair if one of its elements is identical with another element. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
Assertion
Ref Expression
tppreq3  |-  ( B  =  C  ->  { A ,  B ,  C }  =  { A ,  B } )

Proof of Theorem tppreq3
StepHypRef Expression
1 tpeq3 4106 . . 3  |-  ( C  =  B  ->  { A ,  B ,  C }  =  { A ,  B ,  B } )
21eqcoms 2466 . 2  |-  ( B  =  C  ->  { A ,  B ,  C }  =  { A ,  B ,  B } )
3 tpidm23 4119 . 2  |-  { A ,  B ,  B }  =  { A ,  B }
42, 3syl6eq 2511 1  |-  ( B  =  C  ->  { A ,  B ,  C }  =  { A ,  B } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398   {cpr 4018   {ctp 4020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-un 3466  df-sn 4017  df-pr 4019  df-tp 4021
This theorem is referenced by:  tpprceq3  4156  1to3vfriswmgra  25209
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