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Theorem tppreq3 4108
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 4093 . . 3  |-  ( C  =  B  ->  { A ,  B ,  C }  =  { A ,  B ,  B } )
21eqcoms 2441 . 2  |-  ( B  =  C  ->  { A ,  B ,  C }  =  { A ,  B ,  B } )
3 tpidm23 4106 . 2  |-  { A ,  B ,  B }  =  { A ,  B }
42, 3syl6eq 2486 1  |-  ( B  =  C  ->  { A ,  B ,  C }  =  { A ,  B } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437   {cpr 4004   {ctp 4006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-v 3089  df-un 3447  df-sn 4003  df-pr 4005  df-tp 4007
This theorem is referenced by:  tpprceq3  4143  1to3vfriswmgra  25580
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