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Theorem tppreq3 4526
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 4511 . . 3 (𝐶 = 𝐵 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵, 𝐵})
21eqcoms 2786 . 2 (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵, 𝐵})
3 tpidm23 4524 . 2 {𝐴, 𝐵, 𝐵} = {𝐴, 𝐵}
42, 3syl6eq 2830 1 (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  {cpr 4400  {ctp 4402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-v 3400  df-un 3797  df-sn 4399  df-pr 4401  df-tp 4403
This theorem is referenced by:  tpprceq3  4566  1to3vfriswmgr  27688
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