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Theorem en3lplem2VD 38548
Description: Virtual deduction proof of en3lplem2 8457. (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 38258 . . . . . . 7 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   (𝐴𝐵𝐵𝐶𝐶𝐴)   )
2 idn3 38308 . . . . . . 7 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐴   ▶   𝑥 = 𝐴   )
3 en3lplem1VD 38547 . . . . . . 7 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)))
41, 2, 3e13 38443 . . . . . 6 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐴   ▶   𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)   )
54in3 38302 . . . . 5 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))   )
6 3anrot 1041 . . . . . . . . 9 ((𝐴𝐵𝐵𝐶𝐶𝐴) ↔ (𝐵𝐶𝐶𝐴𝐴𝐵))
71, 6e1bi 38322 . . . . . . . 8 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   (𝐵𝐶𝐶𝐴𝐴𝐵)   )
8 idn3 38308 . . . . . . . 8 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐵   ▶   𝑥 = 𝐵   )
9 en3lplem1VD 38547 . . . . . . . 8 ((𝐵𝐶𝐶𝐴𝐴𝐵) → (𝑥 = 𝐵 → ∃𝑦(𝑦 ∈ {𝐵, 𝐶, 𝐴} ∧ 𝑦𝑥)))
107, 8, 9e13 38443 . . . . . . 7 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐵   ▶   𝑦(𝑦 ∈ {𝐵, 𝐶, 𝐴} ∧ 𝑦𝑥)   )
11 tprot 4259 . . . . . . . . . 10 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
1211eleq2i 2696 . . . . . . . . 9 (𝑦 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑦 ∈ {𝐵, 𝐶, 𝐴})
1312anbi1i 730 . . . . . . . 8 ((𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥) ↔ (𝑦 ∈ {𝐵, 𝐶, 𝐴} ∧ 𝑦𝑥))
1413exbii 1772 . . . . . . 7 (∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥) ↔ ∃𝑦(𝑦 ∈ {𝐵, 𝐶, 𝐴} ∧ 𝑦𝑥))
1510, 14e3bir 38434 . . . . . 6 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐵   ▶   𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)   )
1615in3 38302 . . . . 5 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   (𝑥 = 𝐵 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))   )
17 jao 534 . . . . 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 38364 . . . 4 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   ((𝑥 = 𝐴𝑥 = 𝐵) → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))   )
19 3anrot 1041 . . . . . . . 8 ((𝐶𝐴𝐴𝐵𝐵𝐶) ↔ (𝐴𝐵𝐵𝐶𝐶𝐴))
201, 19e1bir 38323 . . . . . . 7 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   (𝐶𝐴𝐴𝐵𝐵𝐶)   )
21 idn3 38308 . . . . . . 7 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐶   ▶   𝑥 = 𝐶   )
22 en3lplem1VD 38547 . . . . . . 7 ((𝐶𝐴𝐴𝐵𝐵𝐶) → (𝑥 = 𝐶 → ∃𝑦(𝑦 ∈ {𝐶, 𝐴, 𝐵} ∧ 𝑦𝑥)))
2320, 21, 22e13 38443 . . . . . 6 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐶   ▶   𝑦(𝑦 ∈ {𝐶, 𝐴, 𝐵} ∧ 𝑦𝑥)   )
24 tprot 4259 . . . . . . . . 9 {𝐶, 𝐴, 𝐵} = {𝐴, 𝐵, 𝐶}
2524eleq2i 2696 . . . . . . . 8 (𝑦 ∈ {𝐶, 𝐴, 𝐵} ↔ 𝑦 ∈ {𝐴, 𝐵, 𝐶})
2625anbi1i 730 . . . . . . 7 ((𝑦 ∈ {𝐶, 𝐴, 𝐵} ∧ 𝑦𝑥) ↔ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))
2726exbii 1772 . . . . . 6 (∃𝑦(𝑦 ∈ {𝐶, 𝐴, 𝐵} ∧ 𝑦𝑥) ↔ ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))
2823, 27e3bi 38433 . . . . 5 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐶   ▶   𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)   )
2928in3 38302 . . . 4 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   (𝑥 = 𝐶 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))   )
30 idn2 38306 . . . . . . 7 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   𝑥 ∈ {𝐴, 𝐵, 𝐶}   )
31 dftp2 4207 . . . . . . . 8 {𝐴, 𝐵, 𝐶} = {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)}
3231eleq2i 2696 . . . . . . 7 (𝑥 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)})
3330, 32e2bi 38325 . . . . . 6 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)}   )
34 abid 2614 . . . . . 6 (𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)} ↔ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶))
3533, 34e2bi 38325 . . . . 5 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)   )
36 df-3or 1037 . . . . 5 ((𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶) ↔ ((𝑥 = 𝐴𝑥 = 𝐵) ∨ 𝑥 = 𝐶))
3735, 36e2bi 38325 . . . 4 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   ((𝑥 = 𝐴𝑥 = 𝐵) ∨ 𝑥 = 𝐶)   )
38 jao 534 . . . 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 38329 . . 3 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)   )
4039in2 38298 . 2 (   (𝐴𝐵𝐵𝐶𝐶𝐴)   ▶   (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))   )
4140in1 38255 1 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383  wa 384  w3o 1035  w3a 1036   = wceq 1480  wex 1701  wcel 1992  {cab 2612  {ctp 4157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-v 3193  df-un 3565  df-sn 4154  df-pr 4156  df-tp 4158  df-vd1 38254  df-vd2 38262  df-vd3 38274
This theorem is referenced by:  en3lpVD  38549
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