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Theorem tppreq3 4238
 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 4223 . . 3 (𝐶 = 𝐵 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵, 𝐵})
21eqcoms 2618 . 2 (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵, 𝐵})
3 tpidm23 4236 . 2 {𝐴, 𝐵, 𝐵} = {𝐴, 𝐵}
42, 3syl6eq 2660 1 (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475  {cpr 4127  {ctp 4129 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-sn 4126  df-pr 4128  df-tp 4130 This theorem is referenced by:  tpprceq3  4276  1to3vfriswmgra  26534  1to3vfriswmgr  41450
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