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Theorem tpid3g 4681
Description: Closed theorem form of tpid3 4682. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by JJ, 30-Apr-2021.)
Assertion
Ref Expression
tpid3g (𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})

Proof of Theorem tpid3g
StepHypRef Expression
1 eqid 2821 . . 3 𝐴 = 𝐴
213mix3i 1332 . 2 (𝐴 = 𝐶𝐴 = 𝐷𝐴 = 𝐴)
3 eltpg 4596 . 2 (𝐴𝐵 → (𝐴 ∈ {𝐶, 𝐷, 𝐴} ↔ (𝐴 = 𝐶𝐴 = 𝐷𝐴 = 𝐴)))
42, 3mpbiri 261 1 (𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3o 1083   = wceq 1538  wcel 2115  {ctp 4544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-tru 1541  df-ex 1782  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-v 3473  df-un 3915  df-sn 4541  df-pr 4543  df-tp 4545
This theorem is referenced by:  tpid3  4682  f1dom3fv3dif  7000  f1dom3el3dif  7001  en3lplem1  9051  en3lp  9053  nb3grprlem1  27148  cplgr3v  27203  cshw1s2  30620  cyc3co2  30789  en3lplem1VD  41334  en3lpVD  41336  limsupequzlem  42157  etransclem48  42717
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