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Theorem en3lplem2VD 41040
 Description: Virtual deduction proof of en3lplem2 9068. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
en3lplem2VD ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem en3lplem2VD
StepHypRef Expression
1 idn1 40770 . . . . . . 7 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   (𝐴𝐵𝐵𝐶𝐶𝐴)   )
2 idn3 40811 . . . . . . 7 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐴   ▶   𝑥 = 𝐴   )
3 en3lplem1VD 41039 . . . . . . 7 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)))
41, 2, 3e13 40944 . . . . . 6 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐴   ▶   𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)   )
54in3 40805 . . . . 5 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))   )
6 3anrot 1094 . . . . . . . . 9 ((𝐴𝐵𝐵𝐶𝐶𝐴) ↔ (𝐵𝐶𝐶𝐴𝐴𝐵))
71, 6e1bi 40825 . . . . . . . 8 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   (𝐵𝐶𝐶𝐴𝐴𝐵)   )
8 idn3 40811 . . . . . . . 8 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐵   ▶   𝑥 = 𝐵   )
9 en3lplem1VD 41039 . . . . . . . 8 ((𝐵𝐶𝐶𝐴𝐴𝐵) → (𝑥 = 𝐵 → ∃𝑦(𝑦 ∈ {𝐵, 𝐶, 𝐴} ∧ 𝑦𝑥)))
107, 8, 9e13 40944 . . . . . . 7 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐵   ▶   𝑦(𝑦 ∈ {𝐵, 𝐶, 𝐴} ∧ 𝑦𝑥)   )
11 tprot 4683 . . . . . . . . . 10 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
1211eleq2i 2908 . . . . . . . . 9 (𝑦 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑦 ∈ {𝐵, 𝐶, 𝐴})
1312anbi1i 623 . . . . . . . 8 ((𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥) ↔ (𝑦 ∈ {𝐵, 𝐶, 𝐴} ∧ 𝑦𝑥))
1413exbii 1841 . . . . . . 7 (∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥) ↔ ∃𝑦(𝑦 ∈ {𝐵, 𝐶, 𝐴} ∧ 𝑦𝑥))
1510, 14e3bir 40935 . . . . . 6 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐵   ▶   𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)   )
1615in3 40805 . . . . 5 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   (𝑥 = 𝐵 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))   )
17 jao 956 . . . . 5 ((𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)) → ((𝑥 = 𝐵 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)) → ((𝑥 = 𝐴𝑥 = 𝐵) → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))))
185, 16, 17e22 40867 . . . 4 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   ((𝑥 = 𝐴𝑥 = 𝐵) → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))   )
19 3anrot 1094 . . . . . . . 8 ((𝐶𝐴𝐴𝐵𝐵𝐶) ↔ (𝐴𝐵𝐵𝐶𝐶𝐴))
201, 19e1bir 40826 . . . . . . 7 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   (𝐶𝐴𝐴𝐵𝐵𝐶)   )
21 idn3 40811 . . . . . . 7 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐶   ▶   𝑥 = 𝐶   )
22 en3lplem1VD 41039 . . . . . . 7 ((𝐶𝐴𝐴𝐵𝐵𝐶) → (𝑥 = 𝐶 → ∃𝑦(𝑦 ∈ {𝐶, 𝐴, 𝐵} ∧ 𝑦𝑥)))
2320, 21, 22e13 40944 . . . . . 6 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐶   ▶   𝑦(𝑦 ∈ {𝐶, 𝐴, 𝐵} ∧ 𝑦𝑥)   )
24 tprot 4683 . . . . . . . . 9 {𝐶, 𝐴, 𝐵} = {𝐴, 𝐵, 𝐶}
2524eleq2i 2908 . . . . . . . 8 (𝑦 ∈ {𝐶, 𝐴, 𝐵} ↔ 𝑦 ∈ {𝐴, 𝐵, 𝐶})
2625anbi1i 623 . . . . . . 7 ((𝑦 ∈ {𝐶, 𝐴, 𝐵} ∧ 𝑦𝑥) ↔ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))
2726exbii 1841 . . . . . 6 (∃𝑦(𝑦 ∈ {𝐶, 𝐴, 𝐵} ∧ 𝑦𝑥) ↔ ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))
2823, 27e3bi 40934 . . . . 5 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐶   ▶   𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)   )
2928in3 40805 . . . 4 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   (𝑥 = 𝐶 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))   )
30 idn2 40809 . . . . . . 7 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   𝑥 ∈ {𝐴, 𝐵, 𝐶}   )
31 dftp2 4625 . . . . . . . 8 {𝐴, 𝐵, 𝐶} = {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)}
3231eleq2i 2908 . . . . . . 7 (𝑥 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)})
3330, 32e2bi 40828 . . . . . 6 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)}   )
34 abid 2806 . . . . . 6 (𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)} ↔ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶))
3533, 34e2bi 40828 . . . . 5 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)   )
36 df-3or 1082 . . . . 5 ((𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶) ↔ ((𝑥 = 𝐴𝑥 = 𝐵) ∨ 𝑥 = 𝐶))
3735, 36e2bi 40828 . . . 4 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   ((𝑥 = 𝐴𝑥 = 𝐵) ∨ 𝑥 = 𝐶)   )
38 jao 956 . . . 4 (((𝑥 = 𝐴𝑥 = 𝐵) → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)) → ((𝑥 = 𝐶 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)) → (((𝑥 = 𝐴𝑥 = 𝐵) ∨ 𝑥 = 𝐶) → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))))
3918, 29, 37, 38e222 40832 . . 3 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)   )
4039in2 40801 . 2 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))   )
4140in1 40767 1 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 843   ∨ w3o 1080   ∧ w3a 1081   = wceq 1530  ∃wex 1773   ∈ wcel 2106  {cab 2802  {ctp 4567 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-v 3501  df-un 3944  df-sn 4564  df-pr 4566  df-tp 4568  df-vd1 40766  df-vd2 40774  df-vd3 40786 This theorem is referenced by:  en3lpVD  41041
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