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Theorem tpid3g 4772
Description: Closed theorem form of tpid3 4773. (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 2737 . . 3 𝐴 = 𝐴
213mix3i 1336 . 2 (𝐴 = 𝐶𝐴 = 𝐷𝐴 = 𝐴)
3 eltpg 4686 . 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 1086   = wceq 1540  wcel 2108  {ctp 4630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-sn 4627  df-pr 4629  df-tp 4631
This theorem is referenced by:  tpid3  4773  f1dom3fv3dif  7288  f1dom3el3dif  7289  en3lplem1  9652  en3lp  9654  tpf  14538  nb3grprlem1  29397  cplgr3v  29452  cshw1s2  32945  cyc3co2  33160  en3lplem1VD  44863  en3lpVD  44865  limsupequzlem  45737  etransclem48  46297
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