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Theorem en3lplem1VD 38100
 Description: Virtual deduction proof of en3lplem1 8394. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
en3lplem1VD ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem en3lplem1VD
StepHypRef Expression
1 idn1 37811 . . . . . . 7 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   (𝐴𝐵𝐵𝐶𝐶𝐴)   )
2 simp3 1056 . . . . . . 7 ((𝐴𝐵𝐵𝐶𝐶𝐴) → 𝐶𝐴)
31, 2e1a 37873 . . . . . 6 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   𝐶𝐴   )
4 tpid3g 4248 . . . . . 6 (𝐶𝐴𝐶 ∈ {𝐴, 𝐵, 𝐶})
53, 4e1a 37873 . . . . 5 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   𝐶 ∈ {𝐴, 𝐵, 𝐶}   )
6 idn2 37859 . . . . . 6 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 = 𝐴   ▶   𝑥 = 𝐴   )
7 eleq2 2677 . . . . . . 7 (𝑥 = 𝐴 → (𝐶𝑥𝐶𝐴))
87biimprd 237 . . . . . 6 (𝑥 = 𝐴 → (𝐶𝐴𝐶𝑥))
96, 3, 8e21 37978 . . . . 5 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 = 𝐴   ▶   𝐶𝑥   )
10 pm3.2 462 . . . . 5 (𝐶 ∈ {𝐴, 𝐵, 𝐶} → (𝐶𝑥 → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶𝑥)))
115, 9, 10e12 37972 . . . 4 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 = 𝐴   ▶   (𝐶 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶𝑥)   )
12 elex22 3190 . . . 4 ((𝐶 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶𝑥) → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))
1311, 12e2 37877 . . 3 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 = 𝐴   ▶   𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)   )
1413in2 37851 . 2 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))   )
1514in1 37808 1 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {ctp 4129 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-sn 4126  df-pr 4128  df-tp 4130  df-vd1 37807  df-vd2 37815 This theorem is referenced by:  en3lplem2VD  38101
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