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Theorem tpid2g 4688
 Description: Closed theorem form of tpid2 4687. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Assertion
Ref Expression
tpid2g (𝐴𝐵𝐴 ∈ {𝐶, 𝐴, 𝐷})

Proof of Theorem tpid2g
StepHypRef Expression
1 eqid 2824 . . 3 𝐴 = 𝐴
213mix2i 1331 . 2 (𝐴 = 𝐶𝐴 = 𝐴𝐴 = 𝐷)
3 eltpg 4604 . 2 (𝐴𝐵 → (𝐴 ∈ {𝐶, 𝐴, 𝐷} ↔ (𝐴 = 𝐶𝐴 = 𝐴𝐴 = 𝐷)))
42, 3mpbiri 261 1 (𝐴𝐵𝐴 ∈ {𝐶, 𝐴, 𝐷})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ w3o 1083   = wceq 1538   ∈ wcel 2115  {ctp 4552 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 2796 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 2803  df-cleq 2817  df-clel 2896  df-v 3481  df-un 3923  df-sn 4549  df-pr 4551  df-tp 4553 This theorem is referenced by:  cplgr3v  27214  cshw1s2  30631  cyc3co2  30800  limsupequzlem  42206
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