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Theorem tppreq3 3464
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 3449 . . 3 (𝐶 = B → {A, B, 𝐶} = {A, B, B})
21eqcoms 2040 . 2 (B = 𝐶 → {A, B, 𝐶} = {A, B, B})
3 tpidm23 3462 . 2 {A, B, B} = {A, B}
42, 3syl6eq 2085 1 (B = 𝐶 → {A, B, 𝐶} = {A, B})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  {cpr 3368  {ctp 3369
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3or 885  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-tp 3375
This theorem is referenced by: (None)
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