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Theorem tpid2g 4718
Description: Closed theorem form of tpid2 4717. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Assertion
Ref Expression
tpid2g (𝐴𝐵𝐴 ∈ {𝐶, 𝐴, 𝐷})

Proof of Theorem tpid2g
StepHypRef Expression
1 eqid 2736 . . 3 𝐴 = 𝐴
213mix2i 1333 . 2 (𝐴 = 𝐶𝐴 = 𝐴𝐴 = 𝐷)
3 eltpg 4632 . 2 (𝐴𝐵 → (𝐴 ∈ {𝐶, 𝐴, 𝐷} ↔ (𝐴 = 𝐶𝐴 = 𝐴𝐴 = 𝐷)))
42, 3mpbiri 257 1 (𝐴𝐵𝐴 ∈ {𝐶, 𝐴, 𝐷})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3o 1085   = wceq 1540  wcel 2105  {ctp 4576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3443  df-un 3902  df-sn 4573  df-pr 4575  df-tp 4577
This theorem is referenced by:  cplgr3v  28032  cshw1s2  31460  cyc3co2  31635  limsupequzlem  43588
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