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Theorem en3lplem2VD 39393
Description: Virtual deduction proof of en3lplem2 8550. (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 39107 . . . . . . 7 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   (𝐴𝐵𝐵𝐶𝐶𝐴)   )
2 idn3 39157 . . . . . . 7 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐴   ▶   𝑥 = 𝐴   )
3 en3lplem1VD 39392 . . . . . . 7 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)))
41, 2, 3e13 39292 . . . . . 6 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐴   ▶   𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)   )
54in3 39151 . . . . 5 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))   )
6 3anrot 1060 . . . . . . . . 9 ((𝐴𝐵𝐵𝐶𝐶𝐴) ↔ (𝐵𝐶𝐶𝐴𝐴𝐵))
71, 6e1bi 39171 . . . . . . . 8 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   (𝐵𝐶𝐶𝐴𝐴𝐵)   )
8 idn3 39157 . . . . . . . 8 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐵   ▶   𝑥 = 𝐵   )
9 en3lplem1VD 39392 . . . . . . . 8 ((𝐵𝐶𝐶𝐴𝐴𝐵) → (𝑥 = 𝐵 → ∃𝑦(𝑦 ∈ {𝐵, 𝐶, 𝐴} ∧ 𝑦𝑥)))
107, 8, 9e13 39292 . . . . . . 7 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐵   ▶   𝑦(𝑦 ∈ {𝐵, 𝐶, 𝐴} ∧ 𝑦𝑥)   )
11 tprot 4316 . . . . . . . . . 10 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
1211eleq2i 2722 . . . . . . . . 9 (𝑦 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑦 ∈ {𝐵, 𝐶, 𝐴})
1312anbi1i 731 . . . . . . . 8 ((𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥) ↔ (𝑦 ∈ {𝐵, 𝐶, 𝐴} ∧ 𝑦𝑥))
1413exbii 1814 . . . . . . 7 (∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥) ↔ ∃𝑦(𝑦 ∈ {𝐵, 𝐶, 𝐴} ∧ 𝑦𝑥))
1510, 14e3bir 39283 . . . . . 6 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐵   ▶   𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)   )
1615in3 39151 . . . . 5 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   (𝑥 = 𝐵 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))   )
17 jao 533 . . . . 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 39213 . . . 4 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   ((𝑥 = 𝐴𝑥 = 𝐵) → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))   )
19 3anrot 1060 . . . . . . . 8 ((𝐶𝐴𝐴𝐵𝐵𝐶) ↔ (𝐴𝐵𝐵𝐶𝐶𝐴))
201, 19e1bir 39172 . . . . . . 7 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   (𝐶𝐴𝐴𝐵𝐵𝐶)   )
21 idn3 39157 . . . . . . 7 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐶   ▶   𝑥 = 𝐶   )
22 en3lplem1VD 39392 . . . . . . 7 ((𝐶𝐴𝐴𝐵𝐵𝐶) → (𝑥 = 𝐶 → ∃𝑦(𝑦 ∈ {𝐶, 𝐴, 𝐵} ∧ 𝑦𝑥)))
2320, 21, 22e13 39292 . . . . . 6 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐶   ▶   𝑦(𝑦 ∈ {𝐶, 𝐴, 𝐵} ∧ 𝑦𝑥)   )
24 tprot 4316 . . . . . . . . 9 {𝐶, 𝐴, 𝐵} = {𝐴, 𝐵, 𝐶}
2524eleq2i 2722 . . . . . . . 8 (𝑦 ∈ {𝐶, 𝐴, 𝐵} ↔ 𝑦 ∈ {𝐴, 𝐵, 𝐶})
2625anbi1i 731 . . . . . . 7 ((𝑦 ∈ {𝐶, 𝐴, 𝐵} ∧ 𝑦𝑥) ↔ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))
2726exbii 1814 . . . . . 6 (∃𝑦(𝑦 ∈ {𝐶, 𝐴, 𝐵} ∧ 𝑦𝑥) ↔ ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))
2823, 27e3bi 39282 . . . . 5 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐶   ▶   𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)   )
2928in3 39151 . . . 4 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   (𝑥 = 𝐶 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))   )
30 idn2 39155 . . . . . . 7 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   𝑥 ∈ {𝐴, 𝐵, 𝐶}   )
31 dftp2 4263 . . . . . . . 8 {𝐴, 𝐵, 𝐶} = {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)}
3231eleq2i 2722 . . . . . . 7 (𝑥 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)})
3330, 32e2bi 39174 . . . . . 6 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)}   )
34 abid 2639 . . . . . 6 (𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)} ↔ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶))
3533, 34e2bi 39174 . . . . 5 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)   )
36 df-3or 1055 . . . . 5 ((𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶) ↔ ((𝑥 = 𝐴𝑥 = 𝐵) ∨ 𝑥 = 𝐶))
3735, 36e2bi 39174 . . . 4 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   ((𝑥 = 𝐴𝑥 = 𝐵) ∨ 𝑥 = 𝐶)   )
38 jao 533 . . . 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 39178 . . 3 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)   )
4039in2 39147 . 2 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))   )
4140in1 39104 1 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383  w3o 1053  w3a 1054   = wceq 1523  wex 1744  wcel 2030  {cab 2637  {ctp 4214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-un 3612  df-sn 4211  df-pr 4213  df-tp 4215  df-vd1 39103  df-vd2 39111  df-vd3 39123
This theorem is referenced by:  en3lpVD  39394
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