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Theorem tpid1g 3643
 Description: Closed theorem form of tpid1 3642. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Assertion
Ref Expression
tpid1g (𝐴𝐵𝐴 ∈ {𝐴, 𝐶, 𝐷})

Proof of Theorem tpid1g
StepHypRef Expression
1 eqid 2140 . . 3 𝐴 = 𝐴
213mix1i 1154 . 2 (𝐴 = 𝐴𝐴 = 𝐶𝐴 = 𝐷)
3 eltpg 3576 . 2 (𝐴𝐵 → (𝐴 ∈ {𝐴, 𝐶, 𝐷} ↔ (𝐴 = 𝐴𝐴 = 𝐶𝐴 = 𝐷)))
42, 3mpbiri 167 1 (𝐴𝐵𝐴 ∈ {𝐴, 𝐶, 𝐷})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ w3o 962   = wceq 1332   ∈ wcel 1481  {ctp 3534 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-3or 964  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-tp 3540 This theorem is referenced by:  rngbaseg  12115  srngbased  12122  lmodbased  12133  ipsbased  12141  ipsscad  12144  topgrpbasd  12151
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