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Theorem tpid2g 4796
Description: Closed theorem form of tpid2 4795. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Assertion
Ref Expression
tpid2g (𝐴𝐵𝐴 ∈ {𝐶, 𝐴, 𝐷})

Proof of Theorem tpid2g
StepHypRef Expression
1 eqid 2734 . . 3 𝐴 = 𝐴
213mix2i 1334 . 2 (𝐴 = 𝐶𝐴 = 𝐴𝐴 = 𝐷)
3 eltpg 4709 . 2 (𝐴𝐵 → (𝐴 ∈ {𝐶, 𝐴, 𝐷} ↔ (𝐴 = 𝐶𝐴 = 𝐴𝐴 = 𝐷)))
42, 3mpbiri 258 1 (𝐴𝐵𝐴 ∈ {𝐶, 𝐴, 𝐷})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3o 1086   = wceq 1537  wcel 2103  {ctp 4652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3484  df-un 3975  df-sn 4649  df-pr 4651  df-tp 4653
This theorem is referenced by:  tpf  14544  cplgr3v  29461  cshw1s2  32919  cyc3co2  33125  limsupequzlem  45577
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