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Theorem en3lplem1VD 44841
Description: Virtual deduction proof of en3lplem1 9650. (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 44572 . . . . . . 7 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   (𝐴𝐵𝐵𝐶𝐶𝐴)   )
2 simp3 1137 . . . . . . 7 ((𝐴𝐵𝐵𝐶𝐶𝐴) → 𝐶𝐴)
31, 2e1a 44625 . . . . . 6 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   𝐶𝐴   )
4 tpid3g 4777 . . . . . 6 (𝐶𝐴𝐶 ∈ {𝐴, 𝐵, 𝐶})
53, 4e1a 44625 . . . . 5 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   𝐶 ∈ {𝐴, 𝐵, 𝐶}   )
6 idn2 44611 . . . . . 6 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 = 𝐴   ▶   𝑥 = 𝐴   )
7 eleq2 2828 . . . . . . 7 (𝑥 = 𝐴 → (𝐶𝑥𝐶𝐴))
87biimprd 248 . . . . . 6 (𝑥 = 𝐴 → (𝐶𝐴𝐶𝑥))
96, 3, 8e21 44728 . . . . 5 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 = 𝐴   ▶   𝐶𝑥   )
10 pm3.2 469 . . . . 5 (𝐶 ∈ {𝐴, 𝐵, 𝐶} → (𝐶𝑥 → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶𝑥)))
115, 9, 10e12 44722 . . . 4 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 = 𝐴   ▶   (𝐶 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶𝑥)   )
12 elex22 3504 . . . 4 ((𝐶 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶𝑥) → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))
1311, 12e2 44629 . . 3 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 = 𝐴   ▶   𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)   )
1413in2 44603 . 2 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))   )
1514in1 44569 1 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  {ctp 4635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-sn 4632  df-pr 4634  df-tp 4636  df-vd1 44568  df-vd2 44576
This theorem is referenced by:  en3lplem2VD  44842
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