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Theorem en3lplem1VD 41957
 Description: Virtual deduction proof of en3lplem1 9121. (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 41688 . . . . . . 7 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   (𝐴𝐵𝐵𝐶𝐶𝐴)   )
2 simp3 1135 . . . . . . 7 ((𝐴𝐵𝐵𝐶𝐶𝐴) → 𝐶𝐴)
31, 2e1a 41741 . . . . . 6 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   𝐶𝐴   )
4 tpid3g 4668 . . . . . 6 (𝐶𝐴𝐶 ∈ {𝐴, 𝐵, 𝐶})
53, 4e1a 41741 . . . . 5 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   𝐶 ∈ {𝐴, 𝐵, 𝐶}   )
6 idn2 41727 . . . . . 6 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 = 𝐴   ▶   𝑥 = 𝐴   )
7 eleq2 2840 . . . . . . 7 (𝑥 = 𝐴 → (𝐶𝑥𝐶𝐴))
87biimprd 251 . . . . . 6 (𝑥 = 𝐴 → (𝐶𝐴𝐶𝑥))
96, 3, 8e21 41844 . . . . 5 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 = 𝐴   ▶   𝐶𝑥   )
10 pm3.2 473 . . . . 5 (𝐶 ∈ {𝐴, 𝐵, 𝐶} → (𝐶𝑥 → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶𝑥)))
115, 9, 10e12 41838 . . . 4 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 = 𝐴   ▶   (𝐶 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶𝑥)   )
12 elex22 3433 . . . 4 ((𝐶 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶𝑥) → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))
1311, 12e2 41745 . . 3 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 = 𝐴   ▶   𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)   )
1413in2 41719 . 2 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))   )
1514in1 41685 1 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {ctp 4529 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-un 3865  df-sn 4526  df-pr 4528  df-tp 4530  df-vd1 41684  df-vd2 41692 This theorem is referenced by:  en3lplem2VD  41958
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