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Theorem tppreq3 4089
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 4074 . . 3  |-  ( C  =  B  ->  { A ,  B ,  C }  =  { A ,  B ,  B } )
21eqcoms 2466 . 2  |-  ( B  =  C  ->  { A ,  B ,  C }  =  { A ,  B ,  B } )
3 tpidm23 4087 . 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 1370   {cpr 3988   {ctp 3990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080  df-un 3442  df-sn 3987  df-pr 3989  df-tp 3991
This theorem is referenced by:  tpprceq3  4122  1to3vfriswmgra  30748
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