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Theorem tpid3g 4777
Description: Closed theorem form of tpid3 4778. (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 2733 . . 3 𝐴 = 𝐴
213mix3i 1336 . 2 (𝐴 = 𝐶𝐴 = 𝐷𝐴 = 𝐴)
3 eltpg 4690 . 2 (𝐴𝐵 → (𝐴 ∈ {𝐶, 𝐷, 𝐴} ↔ (𝐴 = 𝐶𝐴 = 𝐷𝐴 = 𝐴)))
42, 3mpbiri 258 1 (𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3o 1087   = wceq 1542  wcel 2107  {ctp 4633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3954  df-sn 4630  df-pr 4632  df-tp 4634
This theorem is referenced by:  tpid3  4778  f1dom3fv3dif  7267  f1dom3el3dif  7268  en3lplem1  9607  en3lp  9609  nb3grprlem1  28637  cplgr3v  28692  cshw1s2  32124  cyc3co2  32299  en3lplem1VD  43604  en3lpVD  43606  limsupequzlem  44438  etransclem48  44998
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