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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 2735 . . 3 𝐴 = 𝐴
213mix3i 1334 . 2 (𝐴 = 𝐶𝐴 = 𝐷𝐴 = 𝐴)
3 eltpg 4691 . 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 1085   = wceq 1537  wcel 2106  {ctp 4635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-sn 4632  df-pr 4634  df-tp 4636
This theorem is referenced by:  tpid3  4778  f1dom3fv3dif  7288  f1dom3el3dif  7289  en3lplem1  9650  en3lp  9652  tpf  14535  nb3grprlem1  29412  cplgr3v  29467  cshw1s2  32930  cyc3co2  33143  en3lplem1VD  44841  en3lpVD  44843  limsupequzlem  45678  etransclem48  46238
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