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Theorem tpid1g 4730
Description: Closed theorem form of tpid1 4729. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Assertion
Ref Expression
tpid1g (𝐴𝐵𝐴 ∈ {𝐴, 𝐶, 𝐷})

Proof of Theorem tpid1g
StepHypRef Expression
1 eqid 2736 . . 3 𝐴 = 𝐴
213mix1i 1333 . 2 (𝐴 = 𝐴𝐴 = 𝐶𝐴 = 𝐷)
3 eltpg 4646 . 2 (𝐴𝐵 → (𝐴 ∈ {𝐴, 𝐶, 𝐷} ↔ (𝐴 = 𝐴𝐴 = 𝐶𝐴 = 𝐷)))
42, 3mpbiri 257 1 (𝐴𝐵𝐴 ∈ {𝐴, 𝐶, 𝐷})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3o 1086   = wceq 1541  wcel 2106  {ctp 4590
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3447  df-un 3915  df-sn 4587  df-pr 4589  df-tp 4591
This theorem is referenced by:  tpnzd  4741  cplgr3v  28383  cyc3co2  31989  limsupequzlem  43953
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